1 Motion and curve
We shall deal with curves in the Euclidean plane and three dimensional
space. Although we shall need only these two dimensions, we shall start
with the general n-dimensional Euclidean space En.
1.1 1.1. Let I ⊆ R be an open interval. We shall understand points of I as
valus of time t. A map f : I → R could be viewed as a motion whose
trajectory is a curve in a “reasonable” case.
To employ calculus we shall need in particular differentibility of the map
f. We recall the real function ϕ : I → R is of the class Cr if it has continuous
derivatives of the order ≤ r on I. Choosing cartesian coordinates on En,
then f(t) = (f1(t), . . . , fn(t)) is n-tuple of real functions. We say f is of
the class Cr if all fucntions f1, . . . , fn are fo the class Cr.
One needs to show that this notion does not depend on the choice of the
coordinate system. This is indeed true but a direct verification (based on
a transformation from one coordinate sysytem to another one) is difficult.
It is much easier to control independence only on the choice of the origin
of coordinates. The choice of the origin identifies En with its associated
vector space Z(En) which is an n-dimensional Euclidean vector space.
1.2 1.2. We shall therefore focus on the n-dimensional Euclidean vector space
V first. We denote by ||u|| the norm of the vector u and by (u, v) the vector
porduct of vectors u and v.
Definition. A map v : I → V is called a vector function on the interval I.
1.3 1.3. Notion of the limit of a vector function is introduced analogoulsu as
the limit of a real function.
Definition. Vector function v has the limit v0 at the point t0 ∈ I if for
each > 0 there exists δ > 0 such that the following holds: if t = t0 satisfies
|t − t0| < δ then ||v(t) − v0|| < .
We write v0 = limt→t0 v(t).
If v(t0) = limt→t0 v(t) we say the vector function v is continuous at the
point t0.
de1.4 1.4 Definition. If the limit
lim
t→t0
v(t) − v(t0)
t − t0
= lim
t→t0
1
t − t0
v(t) − v(t0) ,
it is called derivative of the vector function v(t) at the piont t0.
1
We shall denote this derivative by dv(t0)
dt or v (t0).
Higher order derivatives are defined by the usual iteration.
1.5 1.5. Let e1, . . . , en is a basis of V . For each t ∈ I we have
v(t) = v1(t)e1 + . . . + vn(t)en.
Real functions vi(t), i = 1, . . . , n are called components of the vector
function v(t).
The following theorem has a simple proof which however belongs to
calculus. Therefore we do not state it.
Theorem. A vector function is continuous if all its components are continuous.
The vector function v(t) has the derivative at the point t0 if all
components have derivative at the point t0. Then it holds
dv(t0)
dt
=
dv1(t0)
dt
, . . . ,
dvn(t0)
dt
.
A similar statement holds also for limits and higher order derivatives.
diffscalar1.6 1.6. Now we shall state an auxiliary result which we shall need later. Let
v(t) and w(t) be two vector functions of the class C1 on I. Their scalar
product v(t), w(t) is a real function of the class C1 on I. Also scalar
products dv
dt , w and v, dw
dt are real function on I.
Theorem. The following holds
d(v, w)
dt
=
dv
dt
, w + v,
dw
dt
.
Proof. In coordinates we have
v(t), w(t) = v1(t)w1(t) + . . . + vn(t)wn(t).
Thus using the chain rule we have
d(v(t), w(t))
dt
=
dv1
dt
w1 + v1
dw1
dt
+ . . . +
dvn
dt
wn + vn
dwn
dt
.
This is the coordinate form of our statement.
1.7 1.7. Let us consider En with its associated vector space V . Let us choose
the origin P ∈ En. Then the map f : I → En determines the vector
function
−→
Pf : I → V ,
−→
Pf(t) =
−−−→
Pf(t) which is called radius (or radious
vector) of f.
2
Definition. The map f : I → En is called motion in the space En. We
say f is motion of the class Cr, if
−→
Pf is a vector function of the class Cr.
Beside the word “motion” we can equivalently say path. The terminology
“motion” is more illustrative, “path” has a more technical nature.
The vector d
−→
Pf
dt does not depend on the choice of the origin. Indeed,
for another point Q ∈ En we have
−→
Pf =
−−→
PQ +
−→
Qf where
−→
Pf is a constant
vector. Thus d
−→
Pf
dt = d
−→
Qf
dt .
de1.8 1.8 Definition. The vector d(
−→
Pf)
dt =: df
dt is called the velocity vector of
the motion f.
It will be also denoted by f .
At the second order we put f = (f ) ; here we already differentiate the
vector function f (and analogously in higher orders).
If f(t) = f1(t), . . . , fn(t) is the coordinate expression of the motion f,
we have
dkf(t)
dtk
=
dkf1(t)
dtk
, . . . ,
dkfn(t)
dtk
.
1.9regularde1.9 1.9 Definition. The motion f : I → En of the class is called regular, if
df(t)
dt = o for every t ∈ I. The point of the paramter t0 at which df(t0)
dt = o
is called a singular point of the motion f.
Here o denotes the zero vector of the space V = Z(En).
We shall show two examples:
(i) In the case of constant motionu f(t) = Q ∈ En, it holds for every
t ∈ I that df(t)
dt = o. Thus for every vaule of time t ∈ I we obtain a singular
point.
(ii) Consider motion x = t2, y = t3 in En, t ∈ (−∞, ∞). This moves
along so colled semicubic parabola y2 − x3 = 0. We have f(t) = (t2, t3),
f (t) = (2t, 3t2) hence f (0) = o. The singular point bod t = 0 is so called
edge, see the picture. JS: missing picture
repar1.10 1.10. Consider another open interval J with the variable τ and a bijective
correspondence ϕ: J → I (i.e. a real function) of the class Cr such that
dϕ
dτ = 0 for every τ ∈ J.
Lemma. If f : I → En is the regular motion of the class Cr then f ◦ϕ: J →
En is also the regular motion of the class Cr.
3
Proof. We have d(f◦ϕ)
dτ = df
dt
dϕ
dτ where dϕ
dτ is a scalar and d(f◦ϕ)
dτ and df
dt are
vectors. Indeed, the coordinate expression of f ◦ϕ is f1(ϕ(τ)), . . . , fn(ϕ(τ)).
The differentiation with respect to τ means to differentiate, at every component,
a composed function with the same inner factor ϕ(τ). Thus d(f◦ϕ)
dτ =
df1
dt
dϕ
dτ , . . . , dfn
dt
dϕ
dτ = df
dt
dϕ
dτ . Since df
dt is a nonzero vector and each dϕ
dt is a
nonzero scalar, also every d(f◦ϕ)
dτ is a nonzero vector.
de1.11 1.11 Definition. Motion f : I → En is called simple, if f is an injective
mapping, i.e. for t1, t2 ∈ I, t1 = t2 we have f(t1) = f(t2).
From the geometric point of view, f is a motion without self-intersections.
simplecurvede1.12 1.12 Definition. The set C ⊂ En is called a simple curve of the classs
Cr, if there is a simple regular motion f : I → En of the class Cr, such that
C = f(I).
The map f : I → En is called parametrization of the simple curve
C. and the map ϕ from 1.10 is called reparametrization of the curve C.
1.13 1.13. Let J be another interval with the variable τ and g: J → En be
another parametrization of the curve C of the class Cr. The rule f ϕ(τ) =
g(τ) deteremines a map ϕ: J → I, t = ϕ(τ), see the picture.
Theorem. ϕ je funkce tdy Cr a plat dϕ
dτ = 0 pro vechna τ ∈ J.
Proof. Considering coordinate epxressions f(t) = f1(t), . . . , fn(t) , g(τ) =
g1(τ), . . . ,
gn(τ) , the functiion ϕ is determined by relations
e1e1 (1) fi(t) = gi(τ) , i = 1, . . . , n .
Consider an arbitrary point τ0 ∈ J and put t0 = ϕ(τ0) ∈ I. Since df(t0)
dt = o,
at least one of components, say k, of this vector is nonzero. Let us write
the relation fk(t) = gk(τ) as
e2e2 (2) fk(t) − gk(τ) = 0 .
The left hand side is a function fo two variables t a τ of the class Cr on the
product I ×J. We have dfk(t0)
dt = 0 hence we can apply the implicit fucntion
theorem to the eqaution (2) Therefore, t is determined as a function of the
class Cr with the variable τ on an open neighbourhood of the point t0. At
every point we find that t = ϕ(τ) is a function of the class Cr. Accoring
4
to the geometric situation, this function satisfies all equations in (1), i.e.
g(τ) = f ϕ(τ) . By differentiation we find euality of vectors dg
dτ = df
dt
dϕ
dτ ,
where dϕ
dτ is a scalar. Here dϕ(t0)
dτ = 0 at some point would mean dg(τ0)
dτ = o,
which is a contradiction with our assumptions.
We have shown that every two of parametrizations of of the class Cr a
simple curve differ by a reparametrization in the sense of 1.10 and 1.12.
globalcurve1.14 1.14. Now we shall introduce the notion of a global curve.
Definition. A subset C ⊂ En is called curve of the class Cr, if at each
point p ∈ C there is its neighbourhood U such that C ∩U is a simple curve
of the class Cr.
A parametrization of the intersection is called local parametrization
of the curve C.
1.15 1.15. Agreement. Henceforth we shall assume the class r of the curve
we consider is sufficiently high for all performed operations. This will not
be usually explicitly stated.
1.16 1.16. We shall she several example in the plane E2.
a) Parabola is a global simple curve. b) Circle is a curve but not a
simple curve. c) This shape “quarterfoil” is a curve in our (i.e. differential
geometric) definition. d) Two circles with the same center can be considered
as one curve (connectivity is not assumed in the definition 1.14. It is in
fact often useful to say that the border of the annulus determined by these
two circles is one curve. JS: fix translation
On the other hand, the whole semicubuc parabola from 1.9, the Descart
curve from in e) or the lemniscata in f) are not curves in our sense. JS: missing picture
de1.17 1.17 Definition. Two parametrizations f(t) and g(τ) of a simple curve
C are called corresponding each other, if dϕ
dτ > 0 where ϕ is the function
from 1.10.
Two congrunet parametrizations determine the same orientation os a
simple curve C. The choice of an orientation of C thus means to determine
the “direction of motion’. This is possible to do in two ways.
de1.18 1.18 Definition. Let f : I → En be a local parametrization of the curve
C v En. The line determined by the point f(t0), t0 ∈ I and the vector
f (t0) is called tangent line of C at the point f(t0).
5
This definition is independent on the choice of parametrization, because
accoridng to 1.10, two different parametrizations determine colinear
vectors. Thus the tangent line at the point f(t0) has a parametrization
f(t0) + v, f (t0) , v ∈ R .
de1.19 1.19 Definition. The deviation of curves C and ¯C at the intersection
point p is the angle of their tangent lines at this point.
contactde1.20 1.20 Definition. We say two curves C and ¯C of the class Cr at the intersection
point p have the contact of the order k, k ≤ r, if there are local
parametrizations f(t) and ¯f(t) of, respectively, of C and ¯C on the same
interval I with f(t0) = ¯f(t0) = p, if
e3e3 (3)
dif(t0)
dti
=
di ¯f(t0)
dti
for all i = 1, . . . , k .
That is, both curves “agree up to the order k” in such parametrization
at the given point.
contactequivpo1.21 1.21 Remark. It is easy to verify that the “contact of the order k of two
curves” is an equivalnce relation.
ve1.22 1.22 Theorem. Two curves C and ¯C have, at the intersection point p,
the contact of the first order, if and only if their tangent lines at the point
p are equal.
Proof. If C and ¯C have the contact of the first order at the point p,
there will exist parametrizations f(t) a ¯f(t), f(t0) = ¯f(t0) = p such that
df(t0)
dt = d ¯f(t0)
dt . Thus their tangent lines are equal. In the opposite direction,
consider ¯C with an arbitrary parametrization ¯f(¯t), f(t0) = ¯f(t0). If both
tangent lines are equal then d ¯f(t0)
d¯t = k df(t0)
dt , k = 0. Let us perform the
reparametrization ¯t = t0 + 1
k (t−t0) of ¯C. Then using the new parametrization
¯f t0 + 1
k (t − t0) of the curve ¯C, we have d ¯f(t0)
dt = d ¯f(t0)
d¯t · d¯t
dt = d ¯f(t0)
d¯t
1
k .
This is equal to df(t0)
dt according to the definition of k. Thus C a and ¯C
have the contact of the first order.
du1.23 1.23 Corollary. Tangent line is the only line which has the contact of the
first order with the given curve.
de1.24 1.24 Definition. The point p ∈ C is called inflection point of the
curve C, if the tangent line at phas the contact of the 2nd order with
the curve C.
6
inflectionve1.25 1.25 Theorem. Let f be a local parametrization of the curve C. Then
p = f(t0) is the inflection point if and only if the vector d2f(t0)
dt2 is colinear
with the vector df(t0)
dt .
Proof. Put v = df(t0)
dt . An arbitrary motion along the tangent line has the
form g(t) = p + h(t)v where h is a real function. We have dg(t0)
dt = dh(t0)
dt v,
d2g(t0)
dt2 = d2h(t0)
dt2 v which are colinear vectors. If C and its tangent line have
the contact of the 2nd order, also vectors df(t0)
dt a d2f(t0)
dt2 are colinear. In
the opposite direction, let d2f(t0)
dt2 = kdf(t0)
dt . Consider the parametrization
of the tangent line
g(t) = p + (t − t0) +
k
2
(t − t0)2
v .
Then dg(t0)
dt = vdf(t0)
dt , d2g(t0)
dt2 = kv = d2f(t0)
dt2 . Thus the curve C has the
contact of the 2nd order with the tangent line.
arclengthde1.26 1.26 Definition. A parameter s of the parametrization f : I → En of the
curve C is called arc-length, if df
ds = 1 for all s ∈ I.
Thus the arc-length denotes “motion with constant norm of the velocity”
along the curve.
Let f(t) be a parametrization of the curve C. We want to find a
reparametrization s = s(t) with the inverse function t = t(s) such that
s is the arc. That is,
1 =
df
ds
=
df
dt
dt
ds
.
Thus ds
dt = df
dt . Assuming parameters s and t are corresponding each
other, we have
ds
dt
=
df
ds
=
df1
dt
2
+ · · · +
dfn
dt
2
.
This means
e4e4 (4) ds = (f1)2 + · · · + (fn)2 dt .
Now we shall find the arc by integration. Thus the arc-length is given on
every simple curve up to an additive constant.
The display (4) shows that our notion of the arc-legth agrees with the
length of the curve as introduced in calculus. It also agrees with the physical
7
meaning in the sense that if we move along a curve with the unique velocity
then the length of the curve agrees with the length of the corresponding
time interval.
arcinflectionve1.27 1.27 Theorem. Assuming f(s) is the arc-length parametrization, f(s0) is
the inflection point if and only if d2f(s0)
ds2 = o.
Proof. The vector df
ds is unit which equivalentnly means
df
ds
,
df
ds
= 1 .
Accoridng to the theorem 1.6, the differentiation of this relation yields
2 df
ds , d2f
ds2 = 0. That is, the vector d2f(s0)
ds2 is perpendicular to the unit vector
df(s0)
ds . These two vector must be colinear at inflection points accoridng to
the theorem 1.25. Thus d2f(s0)
ds2 = o.
ve1.28 1.28 Theorem. The simple curve C where all point are inflection, points
is a part of a line.
Proof. Considering the arc-length parametrization f(s) of the curve C, all
points are inflection points if and only if d2f
ds2 = o. By integration we
obtain df
ds = a for a constant vector a. vektor. One more integration yields
f = as + b where b is another constant vector. This is a parametrization of
a line.
8
2 Plane curves
2.1 2.1. Let us fix cartesian coordinates (x, y) in E2. Parametrization of curves
has the form f(t) = f1(t), f2(t) , df
dt = o. In particularm the graph of teh
function y = f(x), x ∈ (a, b) of the class Cr is a curve of the class Cr. Its
parametrization is g(t) = t, f(t) , t ∈ (a, b), thus dg
dt = 1, df
dt = o. We
term this an explicit expression of a plane curve.
2.2 2.2. Recall that a function f : U → R of two variables defined on an open
set U ⊂ R2 is of the class Cr if it has continuous partial derivatives on U
of all orders ≤ r.
Theorem. Let U ⊂ R2 be an open set and F : U → R be a function of
the class Cr. Assume the set C defined by F(x, y) = 0 is nonempty and
satisfies ∂F(x0, y0): = ∂F(x0,y0)
∂x , ∂F(x0,y0)
∂y = o for all (x0, y0) ∈ C. Then
the curve C is of the class Cr.
Proof. Let F(x0, y0) = 0 and assume that e.g. ∂F(x0,y0)
∂y = 0. Then according
to the implicit function theorem, the set C can be locally expressed in
the form y = f(x) where f(x) is a function of the class Cr. This is a local
parametrization of the curve C. If ∂F(x0,y0)
∂x = 0 we can (again using the
implicit function tehorem) express C locally in the form x = g(y).
Definition. The point (x0, y0) such that F(x0, y0) = 0, ∂F(x0,y0)
∂x = 0,
∂F(x0,y0)
∂y = 0 is called a singular point of the set F(x, y) = 0.
2.3. Examples. (i) Consider the set x2 + y2 = a, a ∈ R, i.e. F(x, y) =pr2.3
x2 + y2 − a. The set F(x, y) = 0 is empty for a < 0 and it is a single point
for a = 0 where both partial derivatives ∂F
∂x = 2x, ∂F
∂y = 2y are zero. Of
course, this point is not a curve. The case a > 0 corresponds to the circle
with the center at the origin and the radius
√
a. The vector ∂F = (2x, 2y)
is then nonzero at all points.
(ii) Consider the Descrat list F(x, y) = x3 + y3 − 3axy = 0. We have
∂F = (3x2 − 3ay, 3y2 − 3ax). Assuming a = 0, (0, 0) is the unique single
point. Assuming a = 0, one easily verifies that the equation ∂F = 0 has
two solutions: (0, 0) and (a, a). The point (a, a) is not on the curve thus
(0, 0) the unique singular point.
(iii) In the case of the semicubic parabola we have F(x, y) = y2 − x3 = 0
m me ∂F = (−3x2, 2y). Thus the origin is the unique singular point.
9
implicittangentve2.4 2.4 Theorem. The tangent line to the curve F(x, y) = 0 at the point(x0, y0)
is given by the equation
(1)
∂F(x0, y0)
∂x
(x − x0) +
∂F(x0, y0)
∂y
(y − y0) = 0 .
Proof. Let f1(t), f2(t) be a parametrization of this curve with f1(t0),
f2(t0) = (x0, y0). Differentiatin of F f1(t), f2(t) = 0 and putting t = t0
we obtain
∂F(x0, y0)
∂x
df1(t0)
dt
+
∂F(x0, y0)
∂y
df2(t0)
dt
= 0 .
The vector ∂F(x0, y0) is thus perpendicular to the vector df(t0)
dt . The equation
(2.4) describes the line through the point (x0, y0) which is perpendicular
to the vector ∂F(x0, y0), i.e. the tangent line.
The condition ∂F(x0, y0) = o for a curve given by the equation, thus
guarantees existence of the tangent line similarly as the condition df(t0)
dt = o
for the parametrization of a curve. There might not be a uniue tangent
line at singular points.
The line through a point on the curve which is perpendicular to the
tangent line at this point is called normal line. The vector ∂F(x0, y0)
thus yields the direction of the normal line.
2.5implparcontact 2.5. According to 1.20, two plane curves C and ¯C have, at a intersection
point p, the contact of the kth order if there exist local parametrizations
f1(t), f2(t) and ¯f1(t), ¯f2(t) of, respectively, C and ¯C on the same interval
I such that
parcontactparcontact (2)
dif1(t0)
dti
=
di ¯f1(t0)
dti
,
dif2(t0)
dti
=
di ¯f2(t0)
dti
, i = 1, . . . , k,
where t0 is the parametr of the intersection point p. The direct approach
to the question whether such parametrizations do or do not exist is rather
complicated in general. However, there is a very simple answer if ¯C is given
by the equation F(x, y) = 0. In this case we can form the one variable
function
(3) Φ(t) = F f1(t), f2(t) .
Theorem. Curves C ≡ f1(t), f2(t) and ¯C ≡ F(x, y) = 0 have, at a
intersection point (x0, y0) = f1(t0), f2(t0) the contact of the kth order if
and only if
implcontactimplcontact (4)
diΦ(t0)
dti
= 0 , i = 1, . . . , k .
10
Proof. Let ¯f1(t), ¯f2(t) be a local parametrization of the curve ¯C such that
the condition (2) for the contact is satisfied. Then
(5) F ¯f1(t), ¯f2(t) = 0
for all t, i.e. all derivatives of the composed function of the left hand side are
zero. Also the function Φ is composed with outer factor F(x, y) and inner
factors f1(t) and f2(t). According to our assumption about the contact,
the derivatives up to the order k of inner factors at the point t0 are the
same for both ¯f1(t) and ¯f2(t). Thus (4) holds.
In the opposite direct, assume (4) holds. Further assume ∂F(x0,y0)
∂y = 0.
We shall locally parametrize the curve ¯C in the form f1(t), g(t) where the
function g(t) is determined by
(6) F f1(t), g(t) = 0 .
This is always possible. Indeed, consider
G(t, y) = F f1(t), y ,
which is well defined on some neighbourhood V of the point (t0, y0). We
have
∂G(t0, v0)
∂y
=
∂F(x0, y0)
∂y
= 0 ,
thus we can use the implicit function theorem for the function G(t, y) = 0.
We need to show that
f2gcontactf2gcontact (7)
dif2(t0)
dti
=
dig(t0)
dti
, i = 1, . . . , k .
On V × R we shall consider the function H of three variabls,
H(t, y, z) = G(t, y) − z .
It satisfies H(t0, y0, 0) = 0 and ∂H(t0,y0,0)
∂y = ∂G(t0,y0)
∂y = 0. Hence using
the implicit function theorem once more, from H = 0 we can locally (and
uniquelly) compute y = K(t, z). Since G t, g(t) = 0 and G t, f2(t) =
Φ(t), we have
g(t) = K(t, 0) a f2(t) = K t, Φ(t) .
Similarly as in the first part of the proof, we have th same outer factor
K(t, z). Derivations of the constant function t → 0 and the function Φ(t)
up to the order k at the point t0 agree (because they are zero). According
to the chain rule, (7) implies (4)
11
2.6 2.6. Now we shall discuss how to approximate an arbitrary plane curve C
at a given point p using circles.
Definition. A circle which has the 2nd order contact with the curve C at
the point p ∈ C, is called osculating circle at the point p.
ve2.7 2.7 Theorem. Assume p ∈ C is not an inflection point. Then there is
exactly one osculating curve at p.
Proof. Denote by (a, b) the center and by r the radius of the circle, i.e. its
equation is
ocircleocircle (8) (x − a)2
+ (y − b)2
− r2
= 0 .
Using the theorem 2.5 we shall find a condition for (8) to have the 2nd
order contact with the curve given by the parametrization f1(t), f2(t) at
the point t0. We have
Φ(t) = f1(t) − a
2
+ f2(t) − b
2
− r2
,
Φ (t) = 2(f1 − a)f1 + 2(f2 − b)f2 ,
Φ (t) = 2(f1)2
+ 2(f1 − a)f1 + 2(f2)2
+ 2(f2 − b)f2 .
Coordinates a, b are solutions of the equation Φ = 0, Φ = 0 which are
equivalent to
osystemosystem (9)
af1 + bf2 = f1f1 + f2f2 ,
af1 + bf2 = f1f1 + f2f2 + f 2
1 + f 2
2 .
Away from inflection points, vectors (f1, f2) and (f1 , f2 ) are linearly independent.
Hence the determinant of the system (9) is nonzero and these
equations determine the unique pair (a, b). The radius r is then given by
the equation Φ = 0.
ve2.8 2.8 Theorem. The radius r of the osculationg curve satisfies
oradiusoradius (10) r2
=
(f 2
1 + f 2
2 )3
(f1f2 − f2f1 )2
Proof. One computes (using e.g. the Cramer’s rule) from (9) that
a = f1 −
f2(f 2
1 + f 2
2 )
f1 f2
f1 f2
, b = f2 +
f1(f 2
1 + f 2
2 )
f1 f2
f1 f2
.
The relation r2 = (f1 − a)2 + (f2 − b)2 then yields (10).
12
2.9 2.9. Osculating curves do not exist at inflection points. The tangent line
has the 2nd order contact with the curve hence this curve would have to
have the contact of the 2nd order with the osculating curve accoring to
1.21. However, a simple computation reveals that a circle has the contact
of the 1st order with its tangent line.
Indeed, we can choose such coordinate system such that the circle is
given by the parametrization x = r cos t, y = r sin t. Its tangent line at the
point t = 0 has the equation x − r = 0. We have Φ(t) = r cos t − r thus
Φ(0) = 0, Φ (0) = −r sin 0 = 0 but Φ (0) = r cos 0 = 0. JS: missing picture
planecurvaturede2.10 2.10 Definition. Let r be the radius of the osculating curve at the point
p ∈ C (away from inflection points). The number κ = 1
r is called curvature
of the curve C vat the point p. We define κ = 0 at inflection
points.
This terminology is motivated by the observation that a smaller radius
of the osculating curve means the curve is more “curved”.
The center of the osculating curve is called center of the curvature
of the curve C at the given point.
de2.11 2.11 Definition. Assume p ∈ C is not the inflection point. If the osculating
curve at p ∈ C has the contact of the 3rd order with C the p is called
vertex of the curve.
We shall show in 2.16 that in the case of an ellipse, our general notion
of vertices agrees with the usual definition of vertices of ellipses.
The oscullating curve at vertex of a curve is called hyperosculating.
arccurvature2.12 2.12. Consider the arc-length parameter s. Then the vector e1 = df
ds is
unit and perpendicular to the vector de1
ds = d2f
ds2 according to 1.26 and the
proof the theorem 1.27. Here the inflection point f(s0) is characterized by
de1(s0)
ds = o.
At the point f(s0) (away from inflection points) we denote by e2(s0)
the unit vector parallel with de1(s0)
ds in the same direction. Thus e1(s0) and
e2(s0) is a pair of orthonormal vectors.
Theorem. We have de1(s0)
ds = κ(s0) and the center of the osculating
circle lies on the halfline given by the point f(s0) and the vector e2(s0).
Proof. One can derive this from the expression for the center of the osculating
curve in (10). But it will be useful for later considerations to perform
13
the whole computation once more (with some simplifications). Since the
osculating circle has the contact of the 1st order with the tangent line, its
center lies on the normal line. Thus this center is of the form f(s0)+re2(s0)
for some r ∈ R. The equation of the circle with this center and the radius
r can be written using the scalar product as
z − f(s0) − re2(s0), z − f(s0) − re2(s0) − r2
= 0 ,
where z = (x, y) is an arbitrary point of the plane. For the computation of
the contact we shall therefore use the function
Φ(s) = f(s) − f(s0) − re2(s0), f(s) − f(s0) − re2(s0) − r2
.
By the differentiation and using 1.6 we obtain
1
2
dΦ
ds
= e1(s), f(s) − f(s0) − re2(s0) .
Conditions Φ(s0) = 0 and dΦ(s0)
ds = 0 are satisfied; geometrically this follows
from the fact that we chose the center on the normal line. One more
differentiaion yields
e2.11e2.11 (11)
1
2
d2Φ
ds2
=
de1(s)
ds
, f(s) − f(s0) − re2(s0) + e1(s), e1(s) .
This must be zero at the point s0 hence
e2.12e2.12 (12) r
de1(s0)
ds
, e2(s0) = 1 .
Since the vector de1(s0)
ds is congruently parallel with the vector e2(s0), the
scalar product (12) is equal to the norm of this vector. Our statement is
thus a direct conseuqence of the definition 2.10.
e12curvaturedu2.13 2.13 Corollary. We have de1(s)
ds = κ(s)e2(s).
Proof. It follows from the theorem 2.12 away from inflection points and
from the theorem 1.27 in inflection points.
ve2.14 2.14 Theorem. We have de2(s)
ds = −κ(s)e1(s).
Proof. Since e2 is a unit vector, we have (e2, e2) = 1. By differentiation
we obtain e2, de2
ds = 0. Thus the vector de2
ds is perpendicular to e2, i.e.
14
de2
ds = ce1. Since vectors e1 and e2 are perpendicular, we have (e1, e2) = 0.
By differentiation we obtain
0 =
de1
ds
, e2 + e1,
de2
ds
= κ + c .
ve2.15 2.15 Theorem. The point f(s0) is the vertex if and only if dκ(s0)
ds = 0.
Proof. We shall continue in the proof of the theorem 2.12 and use also the
corollary 2.13. We obtain
1
2
d2φ
ds2
= κ(s) e2(s), f(s) − f(s0) − re2(s0) + 1 .
Further differentiation yields the condition of the contact of the 3rd order
0 =
1
2
d3φ(s0)
ds3
=
dκ(s0)
ds
· (−r) + κ(s0) − κ(s0)e1(s0), −re2(s0) .
Our statement now follows from e1(s0) ⊥ e2(s0) and r = 0.
vrchol2.16 2.16 Corollary. Consider an arbitrary parametrization f(t) of the curve
C. Then away from inflection points, the point f(t0) is vertex if and only
if dκ(t0)
dt = 0.
Proof. The transformation from t to s is realized using a reparametrization
t = ϕ(s), t0 = ϕ(s0). It follows form the chain rule that
dκ(ϕ(s0))
ds
=
dκ(t0)
dt
dϕ(s0)
ds
.
Here dϕ(s0)
ds = 0 since ϕ is a reparametrization.
From this in particularly follows that vertex of an ellipse in the differential
geometric sense are vertices of an ellipse in the classical sense since
the curavture at these points achieves maximum of minimum.
onlyverticesve 2.17 2.17 Theorem. A simple curve whose every point is vertex, is a part of
the circle.
Proof. The center of the osculating circle is c(s) = f(s) + 1
κ e2(s). If every
point is vertex, κ will be a constant. Thus by differentiation and using
2.13, we obtain
dc(s)
ds
= e1(s) −
1
κ
κe1(s) = o .
Thus c(s) is a fixed point and also the radius 1
κ is constant. All osculating
curves thus coincide and the curve lies on a circle.
15
2.18 Definition. Relations
e2.13e2.13 (13)
df
ds
= e1 ,
de1
ds
= κe2 ,
de2
ds
= −κe1
are called Frenet formulae of the plane curve C without inflection points.
“Moving” frame f(s), e1(s), e2(s) is called Frenet frame of the curve C.
2.19 2.19. Now we shall show how to use relations (13) in order to characterize
congruence of plane curves.
Definition. Curves C and ¯C ⊂ E2 are called congruent if there is an
Euclidean transformation ϕ: E2 → E2 such that ϕ(C) = ¯C.
ve2.20 2.20 Theorem. Let C, ¯C be curves without inflection points, f : I → E2,
¯f : I → E2, respectively their arc-length parametrizations on the same
interval I and κ(s), ¯κ(s), respectively be their curvatures. Then curves C
and ¯C are congruent if and only if κ = ¯κ on I.
Proof. One direction is obvious: an Euclidean transformation maps arclength
to arc-length and preserves the contact thus radii of osculating
circles at corresponding points must be the same. In the opposite direction,
consider C resp. ¯C with Frenet frame f(s), e1(s), e2(s) resp.
¯f(s), ¯e1(s), ¯e2(s) . Thus beside (13) we have also
e2.14e2.14 (14)
d ¯f
ds
= ¯e1 ,
d¯e1
ds
= κ¯e2 ,
d¯e2
ds
= −κ¯e1
with teh same κ. Thus (13) and (14) is the same system of differential
equations for the 6-tuple of real functions which are components of f, e1
and e2. Given s0 ∈ I, the triple of vectors f(s0), e1(s0), e2(s0) as well
as the triple ¯f(s0), ¯e1(s0), ¯e2(s0) is formed by the point and the pair of
orthonormal vectors. Hence there exists a unique Euclidean transformation
ϕ: E2 → E2 which transforms f(s0) to ¯f(s0), e1(s0) to ¯e1(s0) and e2(s0) to
¯e2(s0). Thus the parametrization ¯f : I → E2 of the curve ¯C together with
vector functions ¯e1(s), ¯e2(s) and the parametrization ϕ ◦ f : I → E2 of the
curve ϕ(C) together with vector functions ϕ ◦ e1, ϕ ◦ e2 satisfy the same
system of differential equations with the same initial conditions. According
to the theorem of the unique existence of a solution of a the system of
differential equations, we have ¯f = ϕ ◦ f, ¯e1 = ϕ ◦ e1, ¯e2 = ϕ ◦ e2. The first
relation ¯f = ϕ ◦ f implies ¯C = ϕ(C).
pr2.21 2.21 Example. The assumption that curves C and ¯C are without inflection
points, is essential. Consider curves given explicitly as y = x3 and
16
y = |x|3, x ∈ (−∞, ∞). Both curves are of the class C2 and have the same JS: missing picture
curvature as a function of the arc-length. But they are not congruent.
2.22 2.22. The next statement can be briefly rephrased as that we can prescribe
the curvature arbitrarily.
Theorem. Let κ : I → R be a positive function. Then locally there exists
a curve C parametrized by arc-length on I such that κ is its curvature.
Idea of the proof: We solve the system of equations (13).
Remark. Globally, this curve might not be simple. For example, if κ =
1
r is a constant, the solution of teh corresponding system of differential
equations is the circle x = r cos s
r , y = r sin s
r . Then for s ∈ (−∞, ∞) one
goes along the circle repeatedly.
2.23 2.23. We shall finish this section with a global result about plane curves.
Recall the subset in E2 is called bounded if it lies inside a circle.
Definition. The plane curve C of the class Cr is called oval of the class
Cr if it is the border of a bounded convex set in E2.
Examples: (i) (ii) JS: missing picture
ve2.24 2.24. Four vertex theorem. Each oval of the class C3 without points of
inflection, has at least four vertices.
Proof. Consider C parametrizad by the arc-length f(s) = f1(s), f2(s) on
the interval s ∈ [0, a] such that for s = a the oval closes, i.e. f(0) = f(a).
Thus the curvature κ is in fact defined on the closed interval hence it
reaches its maximum and minimum. This yields two vertices of the oval C.
We can assume that κ ha minimum at s = 0 and it has maximum at some
point b ∈ (0, a). Choose f(0) to be the origin, f(b) on the x-axis and the
orientation of the y-axis such that f2(s) > 0 for s ∈ (0, b). (If this holds
for one point, it holds for all points by the convexity). Then f2(s) < 0 for
s ∈ (b, a) again using the convexity. The case κ(0) = κ(b) i.e. κ equal to
a constant, is the circle (according to Theorem 2.17) and we can exclude
this case. JS: missing picture
Assume now that f(0) and f(b) are unique vertices. Then dκ
ds > 0 on
(0, b) and dκ
ds < 0 on (b, a). The integration by parts now yields
0 <
a
0
dκ
ds
f2ds = κf2
a
0
−
a
0
κ
df2
ds
ds .
17
But κf2
a
0
= 0 because f(0) = f(a) and κ(0) = κ(a).
Let us expand the relation vztah de1
ds = κe2. We have e1 = df1
ds , df2
ds .
Since e2 is the unit vector perpendicular to e1, we have e2 = ± df2
ds , df1
ds
hence d2f1
ds2 = ±κdf2
ds . Thus
0 < −
a
0
κ
df2
ds
ds = ±
a
0
d2f1
ds2
ds = ±
df1
ds
a
0
= 0 ,
because df1(0)
ds = df1(a)
ds according to the periodicity of the parametrization
of the oval. This is a contradiction.
In fact we have shown there exists another point where dκ
ds changes the
sign, i.e. κ has either minimum or maximum at this point. But minima and
maxima appear in pairs. Thus the existence of the fourth vertex follows.
3 Envelope of a family of plane curves
3.1 3.1. Consider the one-parameter family of plane curves given by the equa-
tions
e3.1e3.1 (1) F(x, y, t) = 0 ,
t ∈ I where F(x, y, t) is a function of the class C1 defined on an open
set U ⊂ R3. Let us denote by Ct0 , t0 ∈ I the curve of the equation
F(x, y, t0) = 0, i.e. we consider (1) as the system of plane curves (Ct).
3.2 3.2. Intersection points of curves Ct and Cs, t = s are determined by the
pair of equations
F(x, y, t) = 0 , F(x, y, s) = 0 .
This system of equations is oviously equivalent with the system
F(x, y, t) = 0 ,
F(x, y, s) − F(x, y, t)
s − t
= 0 .
Considering a fixed t, we obtain the following equation in the limit s → t,
e3.2e3.2 (2) F(x, y, t) = 0 ,
∂F(x, y, t)
∂t
= 0 .
Definition. Points determined by the equation (2) are called characteristic
points on the curve Ct. The set of such points for all t ∈ I is called
charakteristic set of the system (Ct).
18
From the computational point of view, we have two basic possibilities
how o express the characteristic set. If we eliminate the parametr t from
(2), we express the characteristic set in the form of an equation G(x, y) = 0.
If we compute x and y from (2) as function of t, we obtain a parametrization
of the charatcteristic set.
3.3 3.3. We say two curves touch each other in the intersection point
if they have the contact of the 1st order, i.e. the same tangent line.
Definition. The curve D with a parametrization f(t), t ∈ (a, b) ⊂ I is
called envelope of the family (Ct) if D touches the curve Ct0 at the point
f(t0) for all t0 ∈ (a, b).
ve3.4 3.4 Theorem. Each envelope of the family (Ct) is a subset of its characteristic
set.
Proof. The condition that each point of the envelope f(t) = f1(t), f2(t)
lies on the curve Ct, is
e3.3e3.3 (3) F f1(t), f2(t), t = 0 .
The condition that tangent lines for D and Ct coincide at the point f(t),
has the form
e3.4e3.4 (4)
∂F(f1(t), f2(t), t)
∂x
df1(t)
dt
+
∂F(f1(t), f2(t), t)
∂y
df2(t)
dt
= 0 .
By differentiation of (3) we obtain
e3.5e3.5 (5)
∂F(f1(t), f2(t), t)
∂x
df1(t)
dt
+
∂F(f1(t), f2(t), t)
∂y
df2(t)
dt
+
∂F(f1(t), f2(t), t)
∂t
= 0 .
Subtracting (4) from (5) yields
e3.6e3.6 (6)
∂F(f1(t), f2(t), t)
∂t
= 0 .
Thus every envelope is a part of the characteristic set.
3.5 3.5. Consider a very simple case of the family of circles centered on the xaxis
and the constant radius r. That is, F(x, y, t) = (x − t)2 + y2 − r2 = 0.
Then ∂F
∂t = −2(x − t) = 0. Putting x = t to the first equation, we get
y = ±r. Of course, both these lines are envelopes.
3.6 3.6. In the opposite direction, we have the following:
19
Theorem. If the curve f(t) is a solution of (2), it is an envelope of the
family (Ct).
Proof. The curve f(t) satisfies (3), hence f(t) ∈ Ct. By differentiation we
obtain (5). Further we have (6) and subtracting (6) from (5), we obtain
(4). Thus f(t) touches the curve Ct.
3.7 3.7. Having a pair of functions x = f1(t), y = f2(t) which is a solution
of (2), then it is an envelope of the family (Ct) assuming further that
f(t) = f1(t), f2(t) is a curve. In particular df
dt = o. The picture shows
first the family of curves centered on a cirle of the radius r with constant
radius < r, where the inner and outer envelopes are cicrles. The second
case is = r, where the inner envelopes “degenerates” to a point.
JS: missing picture
3.8 3.8. Normal lines of an arbitrary plane curves C form a one-parametr family
of curves.
Definition. The characteristic set of the family of normal lines of the curve
C is called evolute of the curve C.
Theorem. Evolute of the curve C without inflection points coincides with
the set of centers of their osculating curves.
Proof. Let z = (x, y) be an arbitrary point in the plane. We shall parametrize
C by the arc-length and consider its Frener frame e1(s), e2(s) at the point
f(s). The equation of the normal line at the point f(s) then is
e3.7e3.7 (7) F(x, y, s) = e1(s), z − f(s) = 0 .
Using Frenet formulae. we find the condition
e3.8e3.8 (8)
∂F
∂s
= κ(s)e2(s), z − f(s) − e1(s), e1(s) = 0 .
The characteristic set is solution of equations (7) and (8), which we shall
find geometrically. It follows from (7) that plyne
z = f(s) + c(s)e2(s) .
Putting this into (8), we get κ(s)c(s) − 1 = 0, thus c(s) = 1
κ(s) . This is the
center of the osculating circle.
20
3.9. Above we found the parametric expression of the evolute,
z(s) = f(s) +
1
κ(s)
e2(s) .
Thus dz
ds = e1(s) − κ (s)
κs2 e2(s) − e1(s). If κ (s) = 0, this vector is nonzero.
This means, that in some neighbourhood of the point, which is not a vertex,
is evolute a cirve.
The picture shows the evolute of an ellipse. Their edges correspond to JS: missing picture
vertices of the ellipse.
4 Spacial curves and surfaces
Beside a parametric expression, a spacial curve can be also given as an
intersection of two surfaces. In the study of spacial curves, we shall use
their contact with certain auxiuliary surfaces. Now we shall give a general
definition of surfaces in E3.
4.1 4.1. We shall need notion of vector functions of two variables. To simplify
the rpesentation, we shall work only with 3-dimensional Euclidean vector
space V .
Coordinates of the point u ∈ R2 will be denoted by (u1, u2). Let D ⊂ R2
be an open set. The mapping w: D → V will be called vector function
of two variables. If e1, e2, e3 is a basis of V , we have w(u) = w(u1, u2) =
w1(u1, u2)e1 + w2(u1, u2)e2 + w3(u1, u2)e3. Real function w1, w2, w3 are
called components of the vector function w and we write
e4.1e4.1 (1) w(u1, u2) = w1(u1, u2), w2(u1, u2), w3(u1, u2) .
The limit and continuity of vector functions of the vector function w are
defined similarly as in 1.3. We say w has the limit v ∈ V at the point u0 =
(u0
1, u0
2), if for each ε > 0 there is δ > 0 such that |u1 −u0
1| < δ, |u2 −u0
2| < δ,
(u1, u2) = (u0
1, u0
2) implies w(u1, u2) − v < ε. We write lim
u→u0
w(u) = v.
Further, w is continuous at the point u0 if lim
u→u0
w(u) = w(u0).
4.2 4.2. Partial derivatives of the vector fucntion w are defined by
∂w(u0)
∂u1
= lim
u1→u0
1
w(u1, u0
2) − w(u0
1, u0
2)
u1 − u0
1
,
∂w(u0)
∂u2
= lim
u2→u0
2
w(u0
1, u2) − w(u0
1, u0
2)
u2 − u0
2
Higher order partial derivatives ∂kw
∂ui
1∂uj
2
, i + j = k, are defined by the usual
iteration.
21
As in 1.5, a vector function is continuous if and only if all its components
are continuos. An anlogousl statement holds also for limits and partial
derivatives. In particular, following (1) we have
∂1w: =
∂w
∂u1
=
∂w1
∂u1
,
∂w2
∂u1
,
∂w3
∂u1
, ∂2w: =
∂w
∂u2
=
∂w1
∂u2
,
∂w2
∂u2
,
∂w3
∂u2
and similarly for higher order partial derivatives.
We say the function w: D → V is of the class Cr, if it has continuous
partial derivatives of the order ≤ r at the point D.
4.3 4.3. Consider V as the associate vector space of E3. Choose an auxiliary
origin P ∈ E3. Then the mapping f : D → E3 determines radius vector
which is the vector function
−→
Pf : D → V ,
−→
Pf(u) =
−−−−→
Pf(u). We put
e4.2e4.2 (2) ∂1f =
∂f
∂u1
=
∂(
−→
Pf)
∂u1
, ∂2f =
∂f
∂u2
=
∂(
−→
Pf)
∂u2
.
Similarly as in 1.7, this does not depend on the choice of the origin P.
Here (2) are vector functions of two variables. By iteration we have
e4.3e4.3 (3) ∂11f =
∂2f
∂u1∂u1
, ∂12f =
∂2f
∂u1∂u2
, ∂22f =
∂2f
∂u2∂u2
and similarly for higher orders.
de4.4 4.4 Definition. The set S ⊂ E3 is called simple surface of the class
Cr, if there is ana open set D ⊂ R2 and an injective mapping f : D → E3
of the class Cr such that S = f(D) and vectors ∂1f a ∂2f are linearly
independent at each point of the set D.
JS: missing picture
We say f is parametrization of the surface S and D is parameter
space.
The condition that vectors ∂1f and ∂2f are linearly indepedent shall
be written in the form ∂1f × ∂2f = o where × denotes the vector product.
We shall illustrate the meaning of this condition on a parametrization of
the plane E3. Consider D = R2 and put
f = P + u1a + u2b , P ∈ E3 , a, b ∈ V, u1, u2 ∈ R .
In coordinates (x, y, z) on E3 we have
x = p1 + u1a1 + u2b1 , y = p2 + u1a2 + u2b2 , z = p3 + u1a3 + u2b3 .
22
Then ∂1f = a a ∂2f = b. From the analytic geometry we know that f
determines the plane if and only if vectors a and b are linearly independent.
If these vector are linearly dependent, we obtain only a line, and the case
a = b = o yiedls only a point.
4.5 4.5. Given the fucntion z = f(x, y) tdy Cr of two variables on D ⊂ R2
then its graph ¯f(x, y) = x, y, f(x) , ¯f : D → R3 is a simple surface of
the class Cr. The reason is that ∂1
¯f = 1, 0, ∂f
∂x , ∂2
¯f = 0, 1, ∂f
∂y and
these vectors are linearly independent everywhere. We say that f(x, y) is
an explicit description of the surface.
de4.6 4.6 Definition. The subset S ⊂ E3 is called surface of the class Cr if
for each p ∈ S there is its neighbourhood U such that U ∩ S is a simple
surface of the class Cr.
Examples. JS: missing picture
a) Rotational paraboloid is globally a simple surface. b) The sphere JS: check the translation
is a surface which is not simple. c) Anuloid is the surface given by the
rotation of the circle around the axis which lies in the same plane and has
empty intersection with the cicrle. The physical model is “pneumatika”. JS: check the translation
d) Also “an v lec” is an interesting global example of a surface.
4.7 4.7. Agreement. Further we shall assume that the class r of the the suface
or function under consideration is high enough for required constructions
and this will not be usually explicitly stated.
4.8 4.8. A curve on a surface will be usually given in the parameter space D,
i.e. u = u(t), tj. u1 = u1(t), u2 = u2(t), t ∈ I. On the surface S = f(D)
then we have the curve f u(t) = f u1(t), u2(t) .
Theorem. Tangent lines of all curves on the surface S at the point pinS
fill the plane which is called tangent plane of the surface S at the point
p.
Proof. Let p = f(u0). The velocity vector of the motion f u(t) , u(t0) = u0
is given by the differentiation of the composed function
e4.4e4.4 (4)
df(u1(t0), u2(t0))
dt
=
∂f(u0)
∂u1
du1(t0)
dt
+
∂f(u0)
∂u2
du2(t0)
dt
.
Hence this is a linear combination of vectors ∂1f(u0) and ∂2f(u0). In
the opposite direction, for arbitrary vector a = a1∂1f(u0) + a2∂2f(u0) we
have the motion u(t) = u1(t), u2(t) such that du1(t0)
dt = a1, du2(t0)
dt = a2.
Considered tangent lines this fill the whole plane given by the point p and
vectors ∂1f(u0) and ∂2f(u0).
23
The tangent plane of the surafce S at the point p will be denoted by
τpS and its associated vector space TpS is called tangent vector space
of S at the point p.
The previous theorem shows the geometric meaning of the condition of
linear independence of vectors ∂1f a ∂2f which guarantees existence of the
tangent plane.
4.9 4.9. Henceforth we fix the coordinate system (x, y, z), i.e. E3 ≈ R3.
Theorem. Let U ⊂ R3 be an open set and F : U → R is a function of
the class Cr such that the set S given by the equation F(x, y, z) = 0 is
nonempty and
∂F(x0, y0, z0): =
∂F(x0, y0, z0)
∂x
,
∂F(x0, y0, z0)
∂y
,
∂F(x0, y0, z0)
∂z
= o
for each (x0, y0, z0) ∈ S. Then S is a surface of the class Cr.
Proof. Let F(x0, y0, z0) = 0 and e.g. ∂F(x0,y0,z0)
∂z = 0. According to the
implicit function theorem the equation F(x, y, z) = 0 locally yiedls z =
f(x, y) for a function f of the class Cr. Locally this is an explicit description
of the surfaces S. If ∂F(x0,y0,z0)
∂y = 0 resp. ∂F(x0,y0,z0)
∂x = 0, one can locally
compute y = g(x, z) resp. x = h(y, z).
The point (x0, y0, z0) at which ∂F(x0, y0, z0) = o is called singular
point of the set F(x, y, z) = 0.
4.10. Examples. (i) The case F(x, y, z) = x2 + y2 + z2 − a is similar as4.10
in ??. The set F(x, y, z) = 0 is empty for a < 0. If a = 0, the equation
is satisfied only by the origin which is the unique singular point. If a > 0,
we have the sphere centered at the origin wit the radius
√
a. The vector
∂F = (2x, 2y, 2z) is nonzero at all points.
(ii) Consider “rotacni kuzel” F(x, y, z) = z2 − x2 − y2 = 0. The point JS: missing picture
(0, 0, 0) is the unique singular point. Observe the tangent plane does not
exist at this point.
ve4.11 4.11 Theorem. The equation of the tangent plane of the surface S given
by the equation F(x, y, z) = 0 at (x0, y0, z0) ∈ S is
e4.5e4.5 (5)
∂F(x0, y0, z0)
∂x
(x−x0)+
∂F(x0, y0, z0)
∂y
(y −y0)+
∂F(x0, y0, z0)
∂z
(z −z0) = 0 .
24
Proof. Let the curve f1(t), f2(t), f3(t) lie on S and goes through the point
bodem (x0, y0, z0) ∈ S for t = t0. Then
F f1(t), f2(t), f3(t) = 0 .
Differentiating of the composed function and putting t = t0, we obtain
e4.6e4.6 (6)
∂F(x0, y0, z0)
∂x
df1(t0)
dt
+
∂F(x0, y0, z0)
∂y
df2(t0)
dt
+
∂F(x0, y0, z0)
∂z
df3(t0)
dt
= 0 .
Thus the normal vector of the plane (5) is perpendicular to the tangent
vector of any curve on S hence (5) is the tangent plane.
de4.12 4.12 Definition. Consider the surface S. The line NpS through the point
p ∈ S and perpendicular to the tangent plane τpS, is called normal line
of the surface S at the point p.
Thus the vector ∂F(x0, y0, z0) is the directional vector of the normal
line of the surface F(x, y, z) = 0 at its point (x0, y0, z0). The condition
∂F = o geometrically guarantees existence of the tangent plane as well as
the condition ∂1f × ∂2f = o in the case of a parametric description of the
surface.
4.13 4.13. We shall study the question when the intersection of two surfaces
e4.7e4.7 (7) F(x, y, z) = 0 , G(x, y, z) = 0
is a curve
Theorem. Let U ⊂ R3 be an open set and F, G: U → R be functions of
the class Cr such that the set C given by the equation (7) is nonempty and
vectors ∂F(x0, y0, z0) and ∂G(x0, y0, z0) are linearly independent for each
(x0, y0, z0) ∈ C. Then C is a curve of the class Cr.
Proof. Since vector ∂F and ∂G are linearly independent, there is at least
one nonzero subdeterminant of the order 2 in the matrix
e4.8e4.8 (8)
∂F
∂x
∂F
∂y
∂F
∂z
∂G
∂x
∂G
∂y
∂G
∂z
.
If this is the subdetereminant
∂F
∂y
∂F
∂z
∂G
∂y
∂G
∂z
,
25
it follows from the generalized implicit function theorem that from (7) one
can locally compute y = f(x) a z = g(x). Here f and g are again function
of the class Cr. (This theorem can be found at 2.6 of the textbook “´Uvod
do glob´aln´ı anal´yzy” which is stated as [5] in the list of references.) Thus
t, f(t), g(t) is locally a parametrization of the curve given by equations
F = 0 and G = 0. If another subdeterminant of the order 2 is nonzero,
we can locally express x and z as fucntions of y or x and y as functions
of z.
4.14 4.14. We shall illustrate the generalized implicit function theorem on tje
simplest example of two linear equations
F(x, y, z) = a1x + a2y + a3z = 0 ,
G(x, y, z) = b1x + b2y + b3z = 0 .
In this case we have
∂F
∂y
∂F
∂z
∂G
∂y
∂G
∂z
=
a2 a3
b2 b3
and if this determinant is nonzer, one can use the Cramer’s rule to compute
y and z.
4.15 4.15. Geometrically, the theorem 4.13 says that the intersection of two
surfaces S1 and S2 is locally a curve in a neighbourhood of a point p ∈
S1 ∩ S2 where the tangent planes τpS1 and τpS2 are different.
A simple examples of two touching spheres (which intersect in a single
point) shows that this condition is necessary.
An interesting example is so called Viviani curve which is the intersection
of the sphere and “v´alce” with half radius which goes through the
JS: add translation
JS: missing picture
center of the sphere, see the view “from above” in a). The tangent planes
of both surfaces are different surfaces with the exception of the point A.
Indeed, here the intersection of both surfaces is not locally a curve in our
sense, see the “front” point of view in b) and the general picture in c).
4.16 4.16. The definition of the contact of a curve with a plane reduces to
contact of two curves.
Definition. We say the curve C and the surface S have the contact of
the k-th order at the intersection point p, if there exists a curve ¯C on S
such that C and ¯C have the contact of the kth order at the point p.
26
One can easily see that C and S have the contact of the 1st order if and
only if the tangent line of the curve lies in the tangent plane of the surface.
4.17 4.17. The following simple condition to determine the contact of a curve
with a surafce is similar to the theorem 2.5. Let S be given by the equation
F(x, y, z) = 0 and C is given by the parametrization f1(t), f2(t), f3(t) .
Theorem. Let f(t0) = (x0, y0, z0) be an intersection point of the curve
C and the surface S. Consider the function Φ(t) = F f1(t), f2(t), f3(t) .
Then C and S have the contact of the order k if and only if
e4.9e4.9 (9)
diΦ(t0)
dti
= 0 , i = 1, . . . , k .
Proof. Let ¯f(t) be a parametrization of the curve ¯C on the surface S such
that derivatives of f(t) and ¯f(t) coincide up to the order k at t = t0. Since
¯C lies on S, we have
e4.10e4.10 (10) F ¯f1(t), ¯f2(t), ¯f3(t) = 0 ,
hence all derivatives with respect to t of the left hand side are zero. The
function Φ(t) and (10) have the same outer factor F(x, y, z) and derivatives
of inner factors up to the order k coincide according to the condition of
contact. Thus (9) holds.
In the opposite direction, let e.g. ∂F(x0,y0,z0)
∂z = 0. According to the
implicit function theorem, the equation
F f1(t), f2(t), z = 0
locally determines the function z = g(t) and the curve ¯C ≡ f1(t), f2(t), g(t)
lies on S. It is sufficient to show
e4.11e4.11 (11)
dif3(t0)
dti
=
dig(t0)
dti
, i = 1, . . . , k .
Put G(t, z) = F f1(t), f2(t), z . This function is defined on some neighborhood
V of the point (t0, z0). Consider the function of three variables
e4.12e4.12 (12) H(t, z, w) = G(t, z) − w .
on V ×R. According to the implicit function theorem, the equation H(t, z, w) =
0 locally allows to compute z = K(t, w). Since G t, g(t) = 0 and G t, f3(t) =
Φ(t), we have
g(t) = K(t, 0) a f3(t) = K t, Φ(t) .
As in the proof of the theorem 2.5, we obtain (11).
27
5 Frenet frame of spacial curves
Consider a curve C ⊂ E3.
ve5.1 5.1 Theorem. There exists a unique plane ω at non-inflection p ∈ C which
has the contact of the 2nd order with C. Then ω is called osculating plane
of the curve C at the point p.
Proof. Consider a parametrization f(t) = f1(t), f2(t), f3(t) of the curve
C and an arbitrary plane ax + by + cz + d = 0. According to 4.17 we
consider the function
Φ(t) = af1(t) + bf2(t) + cf3(t) + d .
For the 2nd order contact we have conditions Φ(t0) = 0 and
dΦ(t0)
dt
= a
df1(t0)
dt
+ b
df2(t0)
dt
+ c
df3(t0)
dt
= 0 ,
d2Φ(t0)
dt2
= a
d2f1(t0)
dt2
+ b
d2f2(t0)
dt2
+ c
d2f3(t0)
dt2
= 0 .
These conditions mean that the normal vector (a, b, c) of the required plane
is tangent to vectors df(t0)
dt and d2f(t0)
dt2 . Since these two vectors are linearly
independent, the reuired plane is unique.
Similarly as the osculating circle of a plane curve, the condition of the
2nd order contact of the osculating plane with the spacial curve means
that the osculating plane approximates the curve in the best way (among
all possible planes).
du5.2 5.2 Corollary. At a non-inflection point f(t0), the associated vector space
of the osculating plane is given by vectors df(t0)
dt a d2f(t0)
dt2 .
Thus its equation can be epxressed in the form
x − f1(t0), y − f2(t0), z − f3(t0)
f1(t0), f2(t0), f3(t0)
f1 (t0), f2 (t0), f3 (t0)
= 0 .
5.3 5.3. Now we can define the following objects at a non-inflection points
p ∈ C: objekty:
(i) The plane ν through the point p perpendicular to the tangent line
is called normal plane.
28
(ii) The intersection n = ν ∩ ω of the normal and osculating planes is
called principal normal line.
(iii) The line b through the point p perpendicular to the osculating plane
is called binormal line.
(iv) The plane determined by the tangent and binormal lines is called
rectifying plane.
JS: missing picture
de5.4 5.4 Definition. Non-inflection point p ∈ C is called planar point if the
osculating plane at this point has the 3rd order contact with the curve C.
ve5.5 5.5 Theorem. The non-inflection point f(t0) is planar if and only if the
vector d3f(t0)
dt3 is linearly independent on vectors df(t0)
dt and d2f(t0)
dt2 .
Proof. Consider the function Φ(t) from the proof of the theorem 5.1 and
two conditions in this proof for the first and second derivatives of Φ(t).
Considering the 3rd order contact, we have moreover
e5.1e5.1 (1)
d3Φ(t0)
dt3
= a
d3f1(t0)
dt3
+ b
d3f2(t0)
dt3
+ c
d3f3(t0)
dt3
= 0 .
Thus the vector d3f(t0)
dt3 lies in the associated vector space of the osculating
plane therefore it is linearly dependent on vectors df(t0)
dt and d2f(t0)
dt2 . In the
opposite direction, if this linear dependence holds then the equation (1) is
a consequence of two equations from the proof of the theorem 5.1. Thus C
has the 3rd order contact with the osculating plane.
5.6 5.6. Let us further consider the arc-length s. We have e1(s) = df(s)
ds which
is the unit vector. Considering a non-inflection point f(s), we denote by
e2(s) the unit collinear with de1(s)
ds in the same direction. Thus
e5.2e5.2 (2)
de1(s)
ds
= κ(s)e2(s) , κ(s) > 0 .
and the vector e2(s) lies in the osculating plane. According to 1.26, the
vector e2(s) is perpendicular to e1(s). Thus e2(s) yields the direction of
the principal normal line at the point f(s).
5.7 5.7. Assume further the space E3 is oriented. By e3(s) we denote the unit
vector perpendicular to e1(s) and e2(s) such that the basis e1(s), e2(s), e3(s)
is positive. Thus the vector e3(s) yields the direction of the binormal.
Definition. The frame f(s0), e1(s0), e2(s0), e3(s0) is called Frenet frame
of the curve C in the non-inflection point f(s0).
29
The argument s will be further usually omitted.
5.8 5.8. Since the vector e2 is unit, by differentiating of the relation (e2, e2) = 1
we obtain e2, de2
ds = 0. Thus
de2
ds
= ce1 + τe3 .
Similarly by differentiating (e1, e2) = 0 we get de1
ds , e2 + e1, de2
ds = 0 i.e.
κ + c = 0. Thus
e5.3e5.3 (3)
de2(s)
ds
= −κ(s)e1(s) + τ(s)e3(s) .
By differentiating (e3, e3) = 1 we see the vectpr de3
ds is perpendicular to e3.
By differentiating of the relation (e1, e3) = 0 we obtain
de1
ds
, e3 + e1,
de3
ds
= 0 .
But de1
ds = κe2 thus the first scalar product is zero, i.e. de3
ds = ke2. Finally
by differentiating (e2, e3) = 0 we get de2
ds , e3 + e2, de3
ds = 0. From this it
follows τ + k = 0 hence
e5.4e5.4 (4)
de3(s)
ds
= −τ(s)e2(s) .
We have shown
Theorem (Frenet formulae). The curve f(s) without inflection points
satisfies
e5.5e5.5 (5)
df
ds = e1,
de1
ds = κe2,
de2
ds = −κe1 +τe3,
de3
ds = −τe2 .
de5.9 5.9 Definition. The number κ(s0) > 0 is called curvature and the number
τ(s0) is called torsion of the spacial curve f(s) in the non-inflection
point f(s0).
5.10 5.10. Frenet formulae of the curve f(s) yield
e5.6e5.6 (6)
df
ds
= e1 ,
d2f
ds2
= κe2 ,
d3f
ds3
=
dκ
ds
e2 + κ(−κe1 + τe3) .
30
Assume 0 ∈ I for the curve f(s), f : I → E3. Thus at s = 0 we have
the Taylor expansion
f(s) = f(0) + s e1(0) +
κ(0)s2
2
e2(0) +
s3
6
dκ(0)
ds
e2(0) − κ2
(0)e1(0)
+ κ(0) τ(0) e3(0) + ν(s) ,e5.7e5.7 (7)
where ν(s) is the vector function whose value and first three derivatives are
zero at the origin. In the other words, we have:
Theorem. Let x, y, z be coordinates with respect to the Frenet frame
f(0), e1(0), e2(0), e3(0) . Then the curve f(s) in a neighbourhood of the
point f(0) is given by
e5.8e5.8 (8)
x = s −
κ2(0)
6
s3
+ ξ(s) ,
y =
κ(0)
2
s2
+
1
6
dκ(0)
ds
s3
+ η(s) ,
z =
κ(0)τ(0)
6
s3
+ ζ(s) ,
where real functions ξ(s), η(s) and ζ(s) have zero value and first three
derivatives at the origin.
Relations (8) yield so called local expansion of the curve f(s) with
respect to its Frenet frame. Using this, we shall study orthogonal
projections of the curve to its three basic planes of the Frenet frame.
5.11. A geometric meaning of the curvature of a plane curve is given by
the definition 2.10. The curve C ≡ f(s) in E3 satisfies
Theorem. Considering a non-inflection point p ∈ C, the curvature of the
curve C is equal to the curvature of its orthogonal projections Cp to the
osculating plane.
Proof. We can assume p = f(0). Accoridng to (7), Cp has parametrization
(or rather its Taylor expansion)
e5.9e5.9 (9) x = s + α(s) , y =
κ(0)
2
s2
+ β(s) ,
where α(s) and β(s) have zero values and first two derivatives at the origin.
The circle
e5.10e5.10 (10) x2
+ y −
1
κ(0)
2
=
1
κ(0)
2
, tj. x2
+ y2
−
2
κ(0)
y = 0
31
has the second order contact with Cp at the origin. Indeed, putting (9) JS: missing picture
into (10) we get
s2
− s2
+ γ(s) = 0 ,
where γ(s) is a function which has zero value and first two derivatives at
the origin.
5.12. Similarly, a parametrization of the orthogonal projection of the curve
C to the normal plane os
y =
κ(0)
2
s2
+
1
6
dκ(0)
ds
s3
+ η(s) , z =
κ(0)τ(0)
6
s3
+ ζ(s) .
Denoting this vector valued function by g(s), we have dg(0)
ds = o. Thus in a
sense, the origin is an edge of the type of semicubic parabola.
JS: missing picture
5.13. The orthogonal projection of C to its rectifying plane is
x = s −
κ2(0)
6
s3
+ ξ(s) , z =
κ(0)τ(0)
6
s3
+ ζ(s) .
Denoting by h(s) this vector valued function, we have dh(0)
ds = (1, 0), d2h(0)
ds2 =
(0, 0). Hence the origin is an inflection point.
JS: missing picture
Comparision of all three projections yields a better understanding how
the curve C moves “through” its Frenet frame.
5.14. It follows from Frenet formulae that planar points of the spacial curve
C are easily characterized in terms of the torsion.
Theorem. The point f(s0) is planar if and only if τ(s0) = 0.
Proof. It follows from (6) that the vector d3f(s0)
ds3 is a linear combination of
e1(s0) a e2(s0) if and only if τ(s0) = 0.
5.15. The curve C ⊂ E3 is called planar if it lies in some plane ⊂ E3.
Since C lies in lies, every point of C is planar hence the torsion of a plane
curve is zero. The opposite direction follows from Frenet formulae.
Theorem. A simple curve where all points are planar, is a part of a plane.
32
Proof. The condition τ = 0 yileds de3
ds = o i.e. e3 is a constant vector.
Consider the plane through the point f(s0) perpendicular the vector e3(s0).
Its equation is e3(s0), w − f(s0) = 0 where w = (x, y, z) is an arbitrary
point in E3. Consider the function ϕ(s) = e3(s0), f(s) − f(s0) . We have
dϕ
ds = e3(s0), e1(s) = 0 since e3(s0) = e3(s). Thus ϕ is a constant function.
Further ϕ(s0) = 0 thus thus the function ϕ is identically zero. The whole
curve lies in the considered plane.
5.16. The basic geometrical meaning of the torsion follows firectly from
(5).
Theorem. It holds |τ| = de3
ds .
Therefore we can say that torsion is the velocity of rotation of the
binormal vector. Zero rotation of course indicates plane curves. GEnerally
one can say that greater absolute value of torison, more the curve diverges
from a plane curve.
5.17 5.17. We shall find a formula for the curvature κ with respect to an
arbitrary parametrization f(t) of the curve C. It follows from (6) that
κ = df
ds × d2f
ds2 . Put t = t(s). The chain rule yields
e5.11e5.11 (11)
df
ds
=
df
dt
dt
ds
,
d2f
ds2
=
d2f
dt2
dt
ds
2
+
df
dt
d2t
ds2
.
Further we know dt
ds = 1 df
dt . Since the vector product of two colinear
vectors is zero, we have
df
ds
×
d2f
ds2
=
df
dt
dt
ds
×
d2f
dt2
dt
ds
2
+
df
dt
d2t
ds2
=
df
dt
×
d2f
dt2
dt
ds
3
.
Thus we have proved
Theorem. It holds
e5.12e5.12 (12) κ =
df
dt × d2f
dt2
df
dt
3
5.18. We shall derive a formula for torsion τ with respect to an arbitrary
parametrization f(t) of the curve C. Recall three vectors u = (u1, u2, u3),
v = (v1, v2, v3), w = (w1, w2, w3) in the oriented three-dimensional Euclidean
vector space determine the exterior product
[u, v, w] =
u1 u2 u3
v1 v2 v3
w1 w2 w3
.
33
Theorem. It holds
e5.13e5.13 (13) τ =
df
dt , d2f
dt2 , d3f
dt3
df
dt × d2f
dt2
2 .
Proof. First we observe the exterior product satisfies
[u, v + au, w + bu + cv] = [u, v, w] .
It follows from (6) that
df
ds
,
d2f
ds2
,
d3f
ds3
= κ2
τ[e1, e2, e3] = κ2
τ ,
as e1, e2, e3 is a positive basis. We shall rewrite (11) (obtained in the proof
of the theorem 5.17) in the form
e5.14e5.14 (14)
df
ds
=
df
dt
dt
ds
,
d2f
ds2
=
d2f
dt2
dt
ds
2
+ g
df
dt
,
where g = d2t
ds2 but we shall not need this fact. Further differentiation yields
e5.15e5.15 (15)
d3f
ds3
=
d3f
dt3
dt
ds
3
+ h
d2f
dt2
+ k
df
dt
,
where we shall not need coefficients h and k explicitly. Thus we have
κ2
τ =
df
ds
,
d2f
ds2
,
d3f
ds3
=
dt
ds
df
dt
,
dt
ds
2 d2f
dt2
,
dt
ds
3 d3f
dt3
=
=
dt
ds
6 df
dt
,
d2f
dt2
,
d3f
dt3
.
Using (12) for κ and the relation dt
ds = 1 df
dt , (13) follows.
5.19 Example. We shall find curvature and torsion of the screw line.
This curve is given by the trajectory of the uniform screw motion. Its JS: missing picture
parametrization therefore is
f(t) = (a cos t, a sin t, bt) , t ∈ (−∞, ∞), a > 0 .
The number a is the radius of the circular cylinder on which the screw line
lies. The number b is called slope of the screw line.
34
Consecutive differentiation yields
f = (−a sin t, a cos t, b) ,
f = (−a cos t, −a sin t, 0) ,
f = (a sin t, −a cos t, 0) .
Thus f × f = (ab sin t, −ab cos t, a2), f × f = a
√
a2 + b2. Further
f =
√
a2 + b2. Podle (12) m me κ = a
a2+b2 . To deteremine torison we
compute the determinant
[f , f , f ] =
−a sin t a cos t b
−a cos t −a sin t 0
a sin t −a cos t 0
= ba2
.
According (13) we have τ = ba2
a2(a2+b2)
= b
a2+b2 .
Thus the screw line has constant curvature and torsion.
5.20 5.20. Recall proper Euclidean transformation in an oriented space
E3 is such Euclidean transformation ϕ: E3 → E3 which preserves orientation.
Similarly as in a plane, we call two curves C, ¯C ⊂ E3 congruent if
there exists such proper Euclidean transformation ϕ such that ϕ(C) = ¯C.
As in the plane, we shall assume that C and ¯C are simple and both are
parametried by the arc-length on the same interval I.
Theorem. Let curves C and ¯C are without inflection points, f : I → E3
and ¯f : I → E3 are their arc-length parametrizationson the same interval I
and κ(s), ¯κ(s) and τ(s), ¯τ(s) are tehir curactures and torsions, respectively.
Then curves C and ¯C are conguent if and only if κ = ¯κ and τ = ¯τ on I.
Proof. On one side, it directly follows from the geometric construction of
the Frenet frame that given two congruent curves, their curvatures and
torsions are the same functions of the arc-length. In the opposite direction,
consider C and ¯C with Frenet frames f(s), e1(s), e2(s), e3(s) and
¯f(s), ¯e1(s), ¯e2(s), ¯e3(s) , respectively. Beside (5), we have also
e5.16e5.16 (16)
d ¯f
ds
= ¯e1 ,
d¯e1
ds
= κ¯e2 ,
d¯e2
ds
= −κ¯e1 + τ ¯e3 ,
d¯e3
ds
= −τ ¯e2
with the same κ and τ. Thus (5) and (16) is the same system of differencial
equations for twelve real functions which are components of f, e1, e2 and
e3. Given s0 ∈ I, both f(s0), e1(s0), e2(s0), e3(s0) as well as ¯f(s0), ¯e1(s0),
35
¯e2(s0), ¯e3(s0) are the point and a positive orthonormal frame. Hence there
is a unique proper Euclidean motion ϕ: E3 → E3 which transforms the first
of these 4-tuples to the second one. Then parametrization ¯f : I → E3 of the
curve ¯C together with vector functions ¯e1, ¯e2 and ¯e3 and parametrizations
ϕ ◦ f : I → E3 of the curve ϕ(C) together with vector functions ϕ ◦ e1,
ϕ ◦ e2 and ϕ ◦ e3 satisfy the same system of differential equations with the
same initial conditions. According to the theorem about unique existence
of solution of a system of differential equations, we in particularly have
¯f = ϕ ◦ f. Thus ¯C = ϕ(C).
5.21 5.21. Similarly as in the plane we also have the opposite direction.
Theorem. Let κ, τ : I → R be real functions, κ > 0. Then locally there
exists a curve C parametrized by the arc-length on I such that κ is its
curvature and τ is its torsion,
5.22 Example. We shall show the screw lines are unique curves with
constant curvature and torsion (Zero torsion corresponds to a circle as the
screw line with zero slope.) Indeed, we computed in (19) that
e5.17e5.17 (17) κ =
a
a2 + b2
, τ =
b
a2 + b2
.
for screw lines. Let κ > 0 and τ be given. Then we compute from (17) that
τ
κ = b
a thus a = kκ, b = kτ for some k > 0. Putting this to the formula for
κ w eget κ = kκ
k2(κ2+τ2)
, i.e. k = 1
κ2+τ2 . It follows from theorems 5.20 and
5.21 that parts of the screw lines with values a = κ
κ2+τ2 and b = τ
κ2+τ2 are
unique curves with given constant κ and τ.
5.23 Remark. Another interesting (however less imporant in practice)
geometrical object determined by the curve C ≡ f(s) is its osculating
sphere. Assuming f(s0) is a non-planar point then there exists a unique
sphere S which has the 3rd order contact with the curve C at f(s0). We
shall only sketch its construction. It follows from the 1st order contact
that the tangent line of the curve at the point f(s0) is also tangent to
S hence the center of the osculating sphere lies on the normal line. Let
f(s0) + ae2(s0) + be3(s0) is this center. We shall write the equation of the
sphere S in the form of scalar product
w − f(s0) − ae2(s0) − be3(s0), w − f(s0) − ae2(s0) + be3(s0) = a2
+ b2
where w = (x, y, z) is an arbitrary point in E3. To determine the contact
of C and S we use the function
Φ(s) = f(s)−f(s0)−ae2(s0)−be3(s0), f(s)−f(s0)−ae2(s0)−be3(s0)−a2
−b2
.
36
Relations Φ(s0) = 0 and dΦ(s0)
ds = 0 hold by construction. Conditions
d2Φ(s0)
ds2 = 0 and d3Φ(s0)
ds3 = 0 yield
e5.18e5.18 (18) a =
1
κ(s0)
, b = −
κ (s0)
κ2(s0)τ(s0)
, κ (s) =
dκ
ds
.
The radius r =
√
a2 + b2 of the usculating sphere thus is
e5.19e5.19 (19) r =
1
κ2|τ|
κ2τ2 +
dκ
ds
2
.
Display relations (18) and (19) illustrate an interesting general akt. According
to theorems 5.20 and 5.21, the curve C is geometrically determined
by its curvature and torsion. Thus also other geometrical objects deteremined
by the curve and its invariants are expressed using κ and τ and their
derivatives with respect to the arc-length.
37
6 The first fundamental form of the surface
Now we start a systematic study of surfaces in E3.
6.1 6.1. Consider a surface S with a local parametric expression f(u1, u2),
(u1, u2) ∈ D, see 4.4. We shall use the abbreviated notation f1 = ∂1f,
f2 = ∂2f. Thus f1(u0), f2(u0) form a basis of the tangent space TpS of the
surface S at the point p = f(u0).
Consider vectors A, B ∈ TpS, A = a1f1 + a2f2, B = b1f1 + b2f2. Their
scalar product is given by
e6.1e6.1 (1) (A, B) = (a1f1 + a2f2, b1f1 + b2f2) .
Put
e6.2e6.2 (2) g11 = (f1, f1) , g12 = (f1, f2) , g22 = (f2, f2) .
That is, gij, i, j = 1, 2 are functions on D. Then (1) can be written in the
form
e6.3e6.3 (3) (A, B) = g11a1b1 + g12(a1b2 + a2b1) + g22a2b2 .
This is a bilinear form on TpS. The corresponding quadratic form deteremines
the length of the vector A,
A = g11a2
1 + 2g12a1a2 + g22a2
2 .
The angle ϕ of vector A, B satisfies
(4) cos ϕ =
g11a1b1 + g12(a1b2 + a2b1) + g22a2b2
g11a2
1 + 2g12a1a2 + g22a2
2 g11b2
1 + 2g12b1b2 + g22b2
2
6.2. Consider a curve u(t) = u1(t), u2(t) on S. We have df
dt = f1
du1
dt +
f2
du2
dt hence
df
dt
= g11
du1
dt
2
+ 2g12
du1
dt
du2
dt
+ g22
du2
dt
2
From the formula for length of an arc of a space curve we have
Theorem. Length s of the arc of the curve u(t) on the surface f(u) between
points with parameters t1 and t2 is
(5) s =
t2
t1
g11
du1
dt
2
+ 2g12
du1
dt
du2
dt
+ g22
du2
dt
2
dt .
38
Thus the differential ds is given by the expression which follows the
symbolem of integral. Its square
e6.6e6.6 (6) (ds)2
= g11(du1)2
+ 2g12du1du2 + g22(du2)2
is a quadratic form corresponding to the bilinear form (3).
6.3 Definition. The quadratic form (6) is called first fundamental form
of the suface. It is denoted by Φ1 or (ds)2.
We shall use the same symbol Φ1 also for the bilinear form determined
by this quadratic form.
6.4 6.4 Example. Consider the sphere S centered at the origin se with the
radius r. Given a point p ∈ S away from the z-axis, consider its projection JS: missing picture
q to the plane (x, y). We shall denote by u1 the angle of the radius vector
of the point q with the positive half x-axis, i.e. u1 ∈ [0, 2π). We denote by
u2 the angle of the radius vector of the point p with the plane (x, y), i.e.
u2 ∈ − π
2 , π
2 . Thus z = r sin u2 and length of the radius vector of q is
r cos u2. Situation in the plane (x, y) corresponds to polar coordinates, i.e.
x = r cos u2 cos u1, y = r cos u2 sin u1. Summarizing, we have
(7) f(u1, u2) = (r cos u1 cos u2, r sin u1 cos u2, r sin u2) ,
u1 ∈ (0, 2π), u2 ∈ − π
2 , π
2 .
The sphere is a not a simple surface hence our parametrization does not
cover half-circle which is intersection of the sphere with the half-plane x ≥ 0
in the (x, z)-plane. However, this incompletness is usually not a problem.
We shall find the first fundamental form of the sphere. We have
f1 = r(− sin u1 cos u2, cos u1 cos u2, 0) ,
f2 = r(− cos u1 sin u2, − sin u1 sin u2, cos u2) .
Thus g11 = (f1, f1) = r2 cos2 u2, g12 = (f1, f2) = 0, g22 = r2. The first
fundamental form of the sphere is of the form
(8) Φ1 = r2
cos2
u2(du1)2
+ (du2)2
.
6.5 6.5. We shall compute the first fundamental form of an explicitly given
surface z = f(x, y), (x, y) ∈ D, see 4.5. Its parametrization is given by
¯f(x, y) = x, y, f(x, y) . Hence ¯f1 = (1, 0, fx), ¯f2 = (0, 1, fy) where fx
and fy are partial derivatives of f with respect to x and y, respectively.
Computing scalar products (2), we obtain
(9) Φ1 = (1 + f2
y ) (dx)2
+ 2fxfy dx dy + (1 + f2
y ) (dy)2
.
39
6.6 Definition. System of curves on a simple surface S is a 1-parameter
family L of curves on S such that there is a unique curve from L through
every point of the surface S.
First consider system L in the parameter space D of the plane (u1, u2).
Assume that tangent lines of curves of the system are not parallel with the
u2-axis. Then system L(u1, u2) of tangent lines of the system L satisfy
e6.10e6.10 (10)
du2
du1
= L(u1, u2) .
We say (10) is differential equation of the system L.
Vector field on the space D is the rule which assignes a vector in
the tangent space TpD for every point p ∈ D. Having a nowhere vanishing
vector field F1(u1, u2), F2(u1, u2) on D tangent to the system L,
non-parllelity with the u2-axis mean F1(u1, u2) = 0. Then the differential
equation of the system L is
(11)
du2
du1
=
F2(u1, u2)
F1(u1, u2)
.
The system L on a surface S is usually given on the parameter space.
Vector field on the surface S is a rule which assignes a vector in the
tangent space TpS to every point p ∈ S.
6.7 Definition. Orthogonal trajectories of the system L on the surface
S is a system L ’ na S such that curves of L and L ’ are perpendicular
at every point.
Theorem. If F1(u1, u2), F2(u,u2) is a coordinate expression of a vector
field tangent to the system L then the differential system of its orthogonal
trajectories is
e6.12e6.12 (12)
du2
du1
= −
g11F1 + g12F2
g12F1 + g22F2
.
Proof. Let (du1, du2) is a tangent vector to the required system L ’. Following
(3), the orthogonality of both systems means
g11F1du1 + g12(F1du2 + F2du1) + g22F2du2 = 0 .
Now (12) follows by an algebraic manipulation.
40
Remark. If there zero in the denominator of the right hand side of (12)
at some point of the surface, it means orthogonal trajectories through this
point are, considered at the parameter space, tangent to the axis u2. Then
we should consider the differential equation of the system with interchanged
axes u1 and u2.
6.8 Example. We shall find orthogonal trajectories for the system u1 +
u2 = konst on the sphere from 6.4. By differentiation be find du1 +du2 = 0,
i.e. the differential equation of this system is du2
du1
= −1. Thus we cna put
F1 = 1 and F2 = −1. We found g11 = r2 cos2 u2, g12 = 0, g22 = r2 an 6.4.
Following (2), the differential equation of orthogonal trajectories is
e6.13e6.13 (13)
du2
du1
= cos2
u2 .
Separating variables in (13) and integrating, we obtain the equation of
orthogonal trajectories in the form
tg u2 = u1 + konst.
6.9 Definition. Net on the surface S are two systems L1, L2 whose
curves have nonzero angle at every point. The net is called orthogonal if
this is the right angle at every point.
A simple example is the parametric coordinate net formed by curves
u1 = konst. and u2 = konst. given by the parametrization f(u1, u2) of the
surface S. The condition f1 ×f2 = o guarantees that angle of curves of two
parametric systems is nonzero.
The following statement will be useful in many specific case.
Theorem. The parametric net is orthogonal if and only if g12 = 0.
Proof. Vectors f1 and f2 are tangent to parametric systems and g12 =
(f1, f2).
6.10 6.10 Lemma. It holds g11g22 − g2
12 > 0.
Proof. The Cauchy inequality says that vectors a, b satisfy |(a, b)| ≤ a b ,
i.e. (a, b)2 ≤ a 2 b 2. Here the equality holds only if these vectors are
collinear. Putting a = f1, b = f2, we have a 2 = g11, b 2 = g22,
(a, b) = g12. Since vectors f1 and f2 are not collinear, the lemma fol-
lows.
41
6.11. A standard result in calculus shows that the area of the surface given
explicitly as z = f(x, y), (x, y) ∈ D (where f is a bounded function on a
bounded space D) is given by the double integral
e6.14e6.14 (14)
D
1 + f2
x + f2
y dx dy
6.12. The map f : D → E3 is bounded if the set f(D) lies in some ball.
Theorem. Let the surface S is given by a bounded map f(u1, u2) on a
bounded space D ⊂ R2. Then its area is given by
e6.15e6.15 (15)
D
g11g22 − g2
12 du1 du2 .
Proof. We know every surface cen be given explicitly in a neighbourhood
of every point and let z = f(x, y) be such parametrization. We have shown
g11 = 1+f2
x, g12 = fxfy, g22 = 1+fy2 in 6.5, hence g11g22−g2
12 = 1+f2
x +f2
y .
Then (15) localy reduces to the usual expresssion redukuje to the usual
expression (14). The global version follows from the additivity of area of
the surface.
The expression dV : = g11g22 − g2
12 du1 du2 is also called volume
element of the surface S. The formula of the area of the surface thus
has the form V =
D
dV.
6.13 Example. We shall find area V of the so called spherical cap on
the sursface with the radius r with the angle α, see the picture. Thus JS: missing picture
D = (0, 2π) × π
2 − α, π
2 . We found g11 = r2 cos2 u2, g12 = r2 in 6.4. Thus
V =
D
r2
cos u2 du1 du2 = r2
2π
0
du1
π/2
π/2 − α
cos u2du2
= 2πr2
sin u2
π/2
π/2 − α
= 2πr2
(1 − cos α) .
The case α = π
2 gives the area 2πr2 of the half of the sphere.
6.14. Summarizing, the first fundamental form Φ1, determined by scalar
products at each tangent space of the surface, is used mainly for computation
of length of curves on surfaces, angle of these curves and area of the
surface. A fundamental theoretical meaning of Φ1 will be discussed later.
42
7 The second fundamental form of the surface
2fundamental
7.1 7.1. Consuider the normal line NpS of the surface S at the point p. We
have two unit vectors on the normal line, a choice of one of them yields
orientation of the line NpS.
Definition. Orientation of the surface S is a choice of orientation of
every normal line in a continuous way.
One can orient every simple surface. Given a parametrization f(u1, u2),
we can chose the direction of the normal as the direction of the vector
product f1 × f2.
7.2. The case of M¨obius strip shows that there surface which cannot be
oriented.
Definition. The surface S, which can be oriented, is called orientable. An
orientable surface together with a choice of orientation is called oriented.
We shall denote the unit vector of the oriented normal line by n. Dependence
on parameters is expressed if we write n(u1, u2). In the case of
orientation determined by the parametrization f(u), we have
e7.1e7.1 (1) n =
f1 × f2
f1 × f2
.
The orthogonality of the normal line and the surface is expressed by equa-
tions
e7.2e7.2 (2) (n, f1) = 0 , (n, f2) = 0 .
7.3 7.3. Further we assume the surface S is oriented.
Given an arbitrary motion γ(t) in the space E3, the vector d2γ
dt2 is
called accelaration. Consider a motion on the surafce S with a local
parametrizaiton f(u) given by u1(t), u2(t) in the parameter space D.
This is the motion γ(t) = f u1(t), u2(t) in E3. Let us compute teh acceleration.
The first derivative of the composed function is given by the
expression
dγ
dt
= f1 u1(t), u2(t)
du1
dt
+ f2 u1(t), u2(t)
du2
dt
.
We shall use an abbreviation
e7.3e7.3 (3) f11 = ∂11f , f12 = ∂12f , f22 = ∂22f .
43
to compute the second derivative. We obtain
e7.4e7.4 (4)
d2γ
dt2
= f11
du1
dt
2
+ 2f12
du1
dt
du2
dt
+ f22
du2
dt
2
+ f1
d2u1
dt2
+ f2
d2u2
dt2
.
Following (2), the scalar product of n and d2γ
dt2 depends only on dγ
dt .
Definition. The scalar product n, d2γ
dt2 is called normal acceleration
corresponding to the vector dγ
dt ∈ TpS. In the case of dγ
dt =1, this is termed
normal curvature of the oriented surface S in the direction of this
vector.
That is, the sign of the normal acceleration depends on orientation fo
the surface.
7.4 7.4. Consider scalar products
e7.5e7.5 (5) h11 = (n, f11) , h12 = (n, f12) , h22 = (n, f22) ,
which are functions on the space D. Following (4), we obtain the rule
which which associates the normal acceleration to every tangent vector
(du1, du2) ∈ TpS. This is the quadratic form on TpS on the space
e7.6e7.6 (6) h11(du1)2
+ 2h12du1 du2 + h22(du2)2
.
Definition. The quadratic form (6) is called second fundamental form
of the surface S and will be denoted by Φ2.
That is, the second fundamental form of the oriented surface S is a rule
which maps every vetor A ∈ TpS to the number Φ2(A) which he obtained
as follows. We consider a motion γ(t) on the surface S such that A = dγ(t0)
dt .
We compute its acceleration d2γ(t0)
dt2 . The number Φ2(A) is then equal to
the scalar product n(γ(t0)), d2γ(t0)
dt2 where n(γ(t0)) is the oriented vector
of the normal line at the point γ(t0).
7.5 7.5. Consider the direction of the nonzero vector A in the tangent space
TpS. The section of the surface S by the plane determined by the normal
line NpS and the direction A is the curve which we call normal section
of the surface in direction A.
The basic geometrical meaningof the form Φ2 is given by
Theorem. The absolute value of the normal curvature in the direction of
the vector A is equal to the curvature of the normal section in this direction.
44
Proof. Consider the parametrization γ(s) of this section by the arc-length,
γ(s0) = p. Then dγ
ds is a unit vector and d2γ(s0)
ds2 is perpendicular to this
vector. We know (form theory of curves) that the norm of d2γ(s0)
ds2 is equal
to the curvature of the normal section. Vectors n(p) and d2γ(s0)
ds2 are thus
colinear. Since n(p) is a unit vector, the absolute value of the scalar product
n(p), d2γ(s0)
ds2 is equal to the norm of the second vector.
7.6. The normal curvature κ in the direction of the vector A = (du1, du2)
satisfies
e7.7e7.7 (7) κ =
h11(du1)2 + 2h12du1 du2 + h22(du2)2
g11(du1)2 + 2g12du1 du2 + g22(du2)2
.
Indeed, the unit vector in this direction is 1
A (du1, du2) where A 2 =
g11(du1)2 + 2g12du1 du2 + g22(du2)2. Substituting this into (6), the display
(7) follows.
7.7 Definition. The point f(u0) ∈ S is called planar point if the form
Φ2(u0) is nonzero, i.e. h11(u0) = 0, h12(u0) = 0, h22(u0) = 0.
7.8 Definition. The surface S is called connected if every two its point
can be connected by a motion which lies in S.
7.9 7.9 Theorem. A simple connected surface S with all points planar is a
part of a plane.
Proof. Pur n1 = ∂1n, n2 = ∂2n. Differentiating (2) with respect to u1 and
u2 yields
e7.8e7.8 (8)
(n1, f1) + (n, f11) = 0 , (n1, f2) + (n, f12) = 0 ,
(n2, f1) + (n, f12) = 0 , (n2, f2) + (n, f22) = 0 .
In particular, this shows that all points on a plane (with constant normal
vector) are planar. Further we use the fact that n is a unit vector. By
differentiating the relation (n, n) = 1, we obtain
e7.9e7.9 (9) (n, n1) = 0 , (n, n2) = 0 .
If every point of the surface S is planar, the second term in every relation
of (8) is zero, cf. (5). Then first two equations of (8) and the first equation
in (9) mean the vector n1 is perpendicular to three lienarly independent
vectors n, f1, f2. Thus n1 is the zero vector. Analogously, it follows from
45
remaining equations of (8) and (9) that n2 is the zero vector. Thus the
normal vector is constant, n = a. Consider the function
ϕ(u1, u2) = a, f(u1, u2) − f(u0
1, u0
2) .
We have ∂ϕ
∂u1
= (a, f1) = 0, ∂ϕ
∂u2
(a, f2) = 0 hence ϕ is a constant function.
Moreover, ϕ(u0
1, u0
2) = 0, i.e. ϕ(u) = 0 for all u. This means that the whole
surface S lies on its tangent plane through the point f(u0).
7.10 Definition. The point f(u0) ∈ S is called spherical point if the
form Φ2(u0) is a constant multiple of the form Φ1(u0).
Thus the spherical point f(u0) is characterized by the condition
e7.10e7.10 (10)
h11(u0) = cg11(u0) , h12(u0) = cg12(u0) , h22(u0) = cg22(u0) , 0 = c ∈ R .
The normal vector n(u) of the sphere centered at the origin with raidus r
satisfies n(u) = 1
r f(u). Equations (8) show that all points on the sphere
are spherical.
7.11 Theorem. A simple connected surface S where all points are spherical,
is a part of a sphere.
Proof. Assume (10) holds. That is,
e7.11e7.11 (11) (n, f11) = c(f1, f1) , (n, f12) = c(f1, f2) , (n, f22) = c(f2, f2) .
It follows from (8) and (11) thast
e7.12e7.12 (12)
(f1, n1+cf1) = 0 , (f1, n2+cf1) = 0 , (f2, n1+cf2) = 0 , (f2, n2+cf2) = 0 .
Further, it follows from (2) and (9) that
e7.13e7.13 (13) (n, n1 + cf1) = 0 , (n, n2 + cf2) = 0 .
Analogously as in the proof of theorem 7.9, it follows from these relations
that
e7.14e7.14 (14) n1 + cf1 = o , n2 + cf2 = o .
By differentiation of the first equation with respect to u2 and the second
equation with resoect to u1, we obtain
(15) n12 +
∂c
∂u2
f1 + cf12 = 0 , n12 +
∂c
∂u1
f2 + cf12 = 0 ,
46
where n12 = ∂2n
∂u1∂u2
. Thus we have the difference
∂c
∂u2
f1 −
∂c
∂u1
f2 = o .
Since vectors f1 and f2 are linearly independent, we have ∂c
∂u1
= 0, ∂c
∂u2
= 0,
i.e. c is a constant. Following (14), the point f + 1
c n is fixed. The distance
of every point on the surface form this point is constant and equal to 1
|c| ,
i.e. S is a part of the corresponding sphere.
de7.12 7.12 Definition. A direction in the tangent plane of the surafce is called
asymptotic direction, if its normal curvature is zero. The tangent line
in this direction is called asymptotic tangent line.
Thus the equation of asymptotic directions is
e7.16e7.16 (16) h11(du1)2
+ 2h12du1 du2 + h22(du2)2
= 0
Every direction is asymptoic in planar points.
Assuming the direction du2 = 0 is not asymptotic, i.e. h11 = 0, put
= du1
du2
. Then (16) yields the quadratic equation for asymptotic directions
e17e17 (17) h11
2
+ 2h12 + h22 = 0 .
Its roots satisfy 1,2 =
−h12±
√
h2
12−h11h22
h11
. Put
e7.18e7.18 (18) h =
h11 h12
h12 h22
= h11h22 − h2
12 .
Thus there are two (real) asymptotic directions for h < 0, both directions
coincide for h = 0 and there are imaginary roots for h > 0. If h11 = 0
and h22 = 0, (16) yields the quadratic equation for the fraction du2
du1
and the
situation is similar. If h11 = 0 and h22 = 0, we have h12 = 0 in a non-planar
point, i.e. asymptotic directions are du1 = 0 and du2 = 0.
de7.13 7.13 Definition. The non-planar point is called hyperbolic or parabolic
or elliptic, if h < 0 or h = 0 or h > 0, respectively.
In spherical points, the inequality 6.10 means h > 0 hence this is a
special case of an elliptic point.
7.14 Definition. The curve C on the surface S is called asymptotic if
its tangent line ate every point is asymptotic tangent line.
47
Thus we have two systems of asymptotic curves on surfaces with only
hyperbolic points, one system of asymptotic curves on surfaces with only
parabolic points and there is no system of asymptotic curves on surfaces
with only elliptic points.
7.15 Theorem. A line in the tangent plane τpS is an asymptoticc line if
and only if it has contact of the second order with the surface.
Proof. If a direction is asymptotic then the normal section in this direction
has zero curvature at the point p. Thus p is inflection point of the normal
section, i.e. the its tangent line has contact of the second order with this
section. In the opposite direction, if a tangent line at the point p ∈ S has
contact of the 2nd order with some curve γ(t) on S, γ(t0) = p, it is the
inflection point of this curve. Thus the vector d2γ(t)
dt2 is colinear with vector
dγ(t0)
dt which is perpendicular to the normal vector n(p), i.e.
e7.19e7.19 (19) n(p),
d2γ(t0)
dt2
= 0 .
7.16. Recall the osculating plane of a spacial curve is not determined in
its inflection points,
Theorem. A curve C on the surface S is asymptotic if and only if, at
each its point, the osculating plane coincides with the tangent plane of the
surface or is not determined.
Proof. The osculating plane of the curve C ≡ γ(t) at the point p = γ(t0) is
determined by vectors dγ(t0)
dt , d2γ(t0)
dt2 if these are linearly independent. Here
dγ(t0)
dt llies in the tangent plane of the surface. Thus the tangent plane of S
coincides with the osculating plane of the curve C if and only if teh normal
vector n(s) is perpendicular to d2γ(t0)
dt2 , i.e. (19) holds. If this is an inflection
point, the vector d2γ(t0)
dt2 is colinear with dγ(t0)
dt and (19) holds as well. In
the opposite direction, if Φ2
dγ(t0)
dt = 0 then (19) holds and similarly as
in the first part of the proof, one verifies this is one of two cases obtained
above.
7.17. We know from 1.28 that a (part of a) line is characterized by the fact
that all its poins are inflection. Thus if there is a (part of a) line on the
surface, it is an asymptotic curve. This yields e.g. asymptotic directions
and curves on regular ruled surfaces, i.e. on the hyperboloid of one sheet
and hyperbolic paraboloid.
48
7.18. Considering hyperbolic points, asymptotic directions divide directions
in the tangent plane to two parts The sign of the normal curvature is JS: missing picture
positive in one of them and negative in the other one. Thus in the positive
part, normal sections lie locally above the tangent plane in the direction of
the oriented normal line and in the negative part, normal section lie on the
opposite part of the tangent plane. Thus the surface lies on both sides of
the tangent plane. A prominent example is the surface z = xy. Axes x and
y lie on this surface, i.e. they are asymptotic curves. The tangent plane at
the origin is z = 0. Assuming x > 0, y > 0 or x < 0, y < 0, the surface
lies above the tangent plane, assuming x > 0, y < 0 nebo x < 0, y > 0, the
surface lies under the tangent plane.
The sign of the curvature in an elliptic point is the same in all directions
hence the whole surface locally lies on one side of the tangent plane. The
simples cases are sphere and ellipsoid.
Another interesting example is the anuloid. On the “out part of the
tire”, the surface lies on one side of the tangent plane, i.e. all points are
elliptic there. The the whole inner part of the anuloid lies locally lies on
both sides of the tangent plane, i.e. these are hyperbolic points. “Bottom
and top” circles are formed by parabolic points.
7.19 Remark. Finally we shall show how one can characterize planar and
spherical point using the notionof contact of surfaces.
Let p be a common point of S and ¯S. We say surfaces S and ¯S have
contact of the order k at the point p if for every curve C ⊂ S through
the point p there exists a curve ¯C ⊂ ¯S such that curves C and ¯C contact of
the kth order at the point p. It is shown in [5] that this is an equivalence
relation and also a computational criterion (similar to the case of curves
and surfaces in 2.5 and 4.7) is derived.
Assuming the surface S is given by a parametrization f(u) and the
surface ¯S is given by an equation F(x, y, z) = 0 then we shall form a
function of two variables
Φ(u1, u2) = F f1(u1, u2), f2(u1, u2), f3(u1, u2) .
It holds that if surfaces S and ¯S have contact of the kth order at the joint
point p = f(u0) if and only if all partial derivatives of the function Φ at
the point u0 = (u0
1, u0
2) up to the order ≤ k are zero. The case k = 1 then
means two surfaces have contact of the 1st order at a joint pont if and only
if they have the same tangent line at this point.
49
Considering the surface ¯S,
ax + by + cz + d = 0 ,
we have
Φ(u1, u2) = af1(u1, u2), bf2(u1, u2), cf3(u1, u2) + d .
Conditions for the contact of the first order
a∂1f1(u0)+b∂1f2(u0)+c∂1f3(u0) = 0 , a∂2f1(u0)+b∂2f2(u0)+c∂2f3(u0) = 0
mean that the vector (a, b, c) is colinear with the normal vectorem n(u0)
of the surface S at the point f(u0). The condition for contact of the 2nd
order is
n(u0), ∂11f(u0) = 0 , n(u0), ∂12f(u0) = 0 , n(u0), ∂22f(u0) = 0 .
Thus the point p ∈ S is planar if and only if the tangent plane at this point
has contact of the 2nd with the surface.
A similar computation shows that the point f(u0) ∈ S is spherical if
and only if there exists a sphere Q such that S and Q have contact of the
2nd order at the point f(u0).
50
8 Principal curves
principal
8.1. Values of the normal curvature of the surface S ≡ f(u) at its nonplanar
point p can be visualize in the folloing way. We consider the segment
of the length 1√
|κ|
on the tangent line (in both directions) in a nonasymptotical
direction. Here 1√
|κ|
denotes the normal curvature in this direction.
If a1f1(p) + a2f2(p) is such a vector, square of its norm is 1
|κ| , i.e.
e8.1e8.1 (1) g11a2
1 + 2g12a1a2 + g22a2
2 =
1
|κ|
.
But κ is given by 7.(7) hence (1) is equivalent to rovnici
e8.2e8.2 (2) |h11a2
1 + 2h12a1a2 + h22a2
2| = 1 .
Definition. The curve (2) is called Dupin indikatrix at a non-planar
point of the surface.
8.2 8.2. The curve (2) is an ellipse in elliptic points. Considering the equation
of the unit circle in our affine coordinates in our tangent plane is
g11a2
1 + 2g12a1a2 + g22a2
2 = 1
then it follows from 7.(10) that the ellipse is a circle precisely in spherical
points of the surface.
Considering a hyperbolic point and changing suitable coordinates, we
can transform the equation h11a2
1 + 2h12a1a2 + h22a2
2 = 1 to the form
e8.3e8.3 (3)
x2
a2
−
y2
b2
= 1 .
Thus (2) corresponds to a pair of so called conjugated hyperbolas which JS: missing picture
is formed by (3) and the hyperbola x2
a2 − z2
b2 = −1.
Considering a parabolic point, (2) is a pair of parallel lines in the tangent
plane with the point of the surface in the middle. Indeed, we have h11h22 =
h2
12 in this case. Considering h11 > 0, h12 > 0, we have h12 = ±
√
h11
√
h22.
Consider the case of the positive sign first. Then the equation (2) has the
form
e8.4e8.4 (4) 1 = h11a2
1 + 2 h11 h22a1a2 + h22a2
2 = h11a1 + h22a2
2
.
This is an equation of the pair of parallel lines
(5) 1 = h11a1 + h22a2 , −1 = h11a1 + h22a2
51
with the origin in the middle. The case of the negative sign yields the sema
result. A similar computation yields the same result for h11 < 0, h22 < 0.
8.3. Assuming a non-spherical point, we define axes of the Dupin indikatrix
as axes of the ellipse or common axes of the pair of conjugated JS: missing picture
hyperbolas or as the axis of the pair of parallel lines together with the
parpendicular line through the origin.
Definition. Directions of the Dupin indikatrix are called principal directions
of the surface S at a given point. A curve on S which touches
a principal direction at every point is called principal curve.
Principal directions are not defined in planar and spherical points.
We thus have the net of principal curves on surfaces without planar and
spherical points. This net is orthogonal.
8.4 8.4. Since Φ2 is a quadratic form, it deteremines the polar bilinear form
denoted by the same symbol. Given two vectors A = (a1, a2), B = (b1, b2) ∈
TpS, we have
e8.6e8.6 (6) Φ2(A, B) = h11(p)a1b1 + h12(p)(a1b2 + a2b1) + h22(p)a2b2 .
The condition Φ2(A, B) = 0 depends only on directions of the vectors A,
B. This is the condition of polar conjugation with respect to Φ2(p).
Definition. Directions in the tangent plane determined by nonzero vectors
A, B ∈ TpS are called conjugated if they are polar conjugated with respect
to Φ2(p).
From the computational point of view, the condition of conjugation is
given by (6) beeing zero.
ve8.5 8.5 Theorem. Principal directions of the surface are directions which are
in the same time conjugated and perpendicular.
Proof. We know from analytic geometry that this characterizes axes ellipses
and hyperbolas. The case of parallel lines can be computed analogously.
8.6 8.6. Beside (6) beeing zero, principal directions satisfy also the condition
of orthogonality
e8.7e8.7 (7) Φ1(A, B) = g11a1b1 + g12(a1b2 + a2b1) + g22a2b2 = 0 .
52
If (b1, b2) is a nonzero direction satisfying (7) and for which (6) is zero, we
have a system of two homogeneous linear equations with a nonzero solution.
The determinant of the system is zero,
e8.8e8.8 (8)
g11a1 + g12a2, g12a1 + g22a2
h11a1 + h12a2, h12a1 + h22a2
= 0 .
Paasing to the diferentials du1 = a1, du2 = a2, we get
Theorem. The differential equation of the net of principal curves is
e8.9e8.9 (9)
g11du1 + g12du2, g12du1 + g22du2
h11du1 + h12du2, h12du1 + h22du2
= 0 .
Since (9) is generally a quadratic equation for th fraction du2
du1
. Its two
solutions du2
du1
= F1(u1, u2), du2
du1
= F2(u1, u2) are diferential equations of
these two systems of principal curves.
de8.7 8.7 Definition. Normal curvatures κ1, κ2 in principal directions are called
principal curvatures of the surface. The sum H = κ1 +κ2 of principal
curvatures is called mean curvature, the product K = κ1κ2 is called
Gauss (or total) curvature.
Considering spherical points, the normal curvature has the same value
κ. Here we define H = 2κ, K = κ2. Considering planar points, all normal
curvatures are zero. Here we put H = 0, K = 0.
A change of orientation of the surface S changes the sign of the normal
curvature. The sign of the mean curvature H thus depeneds on the
orientation of the surface, however the sign of the Gauss curvature K is
independent on the orientation of the surface.
8.8 8.8. The Dupin indikatrix shows that the normal curvature has extremals
at principal directions. We shall use that to derive a formula for principal
curvatures. The following computation concerns only the “generic” case but
one can show that the result holds in all cases. Considering the direction
= du1
du2
then according to 7.(7), the normal curvature κ( ) in this direction
satisfies
κ( ) =
h11
2 + 2h12 + h22
g11
2 + 2g12 + g22
.
To simplify the computation, we shall write this in the form
e8.10e8.10 (10) κ(g11
2
+ 2g12 + g22) − (h11
2
+ 2h12 + h22) = 0 .
53
Differentiating this with respect to and using the condition fors extremals
dκ
d = 0, we get
e8.11e8.11 (11) κ(g11 + g12) − (h11 + h12 = 0 .
Multiplying this by to − and adding the result to (10), we odtain
e8.12e8.12 (12) κ(g12 + g22) − (h12 + h22) = 0
Putting back = du1
du2
and using an algebraic manipulation, (11) and (12)
have the form
e8.13e8.13 (13)
(κg11 − h11) du1 + (κg12 − h12) du2 = 0
(κg12 − h12) du1 + (κg22 − h22) du2 = 0 .
Here (du1, du2) is a nonzero direction which realizes the extremal. Thus
the determinant of the system of two linear equations (13) must be zero.
Therefore
Theorem. Principal curvatures κ1, κ2 are roots of the quadratic equation
e8.14e8.14 (14)
κg11 − h11, κg12 − h12
κg12 − h12, κg22 − h22
= 0 .
8.9 8.9. A simple corollary of (14) is
Theorem. The mean and the Gauss curvature satisfy
e8.15e8.15 (15) H =
g11h22 − 2g12h12 + g22h11
g11g22 − g2
12
, K =
h11h22 − h2
12
g11g22 − g2
12
.
Proof. It follows from (14) that
κ2
(g11g22 − g2
12) − κ(g11h22 − 2g12h12 + g22h11) + (h11h22 − h2
12) = 0 .
The sum H = κ1 + κ2 and the product K = κ1κ2 of roots have the form
(15) using a well known properties of of quadratic equations. kvadratick
rovnice.
Further we shall show that (15) holds also in spherical and planar points.
According to 7.(7), a spherical point satisfies hij = κgij, i = 1, 2 where κ is
a common value of the normal curvature in all directions. Then (15) means
H = 2κ, K = κ2. We have hij = 0 in planar points, i.e. H = 0 and K = 0.
54
8.10 8.10. Since g11g22 − g2
12 > 0 and h11h22 − h2
12 is the expression used in the
definition 7.13, we obtained a new insight in this definition.
Corollary. Elliptic, parabolic and hyperbolic pointsare characterized by
the condition K > 0, K = 0 and K < 0, respectively.
Remark. Since K = 0 in planar points, planar points are sometimes considered
as parabolic points.
pr8.11 8.11 Example. The Gauss curvatureof the sphere with radius r is 1
r2 .
Indeed, all its points are spherical and the normal section in every direction
is a circle with radius r. Thus K = 1
r2 .
8.12 8.12. The following formula nicely describes the normal curvature in an
arbitrary direction using principal curvatures.
Theorem (Euler formula). Let σ1 and σ2 are principal directions at the
point p of the surface S, let κ1 and κ2 are corresponding principal curvatures
and let s be the direction which has the angle σ1 with ϕ. Then the
normal curvature κs in this direction satisfies
e8.16e8.16 (16) κs = κ1 cos2
ϕ + κ2 sin2
ϕ .
Proof. Let e1, e2 be unit vectors in directions σ1, σ2, respectively. We can
consider parameters u1, u2 on S such that e1 and e2 are tangent vectors of
the parametric net, i.e. e1 = (du1, 0), e2 = (0, du2). Then g11(p) = g22(p) =
1, g12(p) = 0 and conjugacy of directions σ1 and σ2 yields h12(p) = 0. It
follows from the general formula 7.(7) for κ that κ1 = h11(p), κ2 = h22(p).
The unit vector in the direction s has the form e1 cos ϕ + e2 sin ϕ. Putting
this vector into 7.(7), we get κs = κ1 cos2 ϕ + κ2 sin2
ϕ.
8.13 8.13. We shall discuss one more geometric property which directly characterizes
principal curves. Given a curve γ(t) on the surface S, we denote by
nγ the 1-parameter system of normal vectors along γ.
Theorem. The curve γ(t) is a principal curve of the surface S if and only
if the vector
dnγ
dt is colinear with the vector dγ
dt for all t.
Proof. Let S be given by the parametrization f(u) and γ is given by
u1(t), u2(t) in the parameter space. That is,
e8.17e8.17 (17)
dγ
dt
= f1
du1
dt
+ f2
du2
dt
.
55
Assume the vector
e8.18e8.18 (18) a1f1 + a2f2
is perpendicular to (17). Similarly we have nγ(t) = n u1(t), u2(t) hence
e8.19e8.19 (19)
dnγ
dt
= n1
du1
dt
+ n2
du2
dt
.
This vector lies in the tangent plane since vectors n1 and n2 are perpendicular
to n, see 7.(9). Vectors (17) and (19) are colinear if and only if vectors
(18) and (19) are perpendicular. Using the formula 7.(8), we get
0 = f1a1 + f2a2, n1
du1
dt
+ n2
du2
dt
= − h11a1
du1
dt
+ h12 a2
du1
dt
+ a1
du2
dt
+ h22a2
du2
dt
.e8.20e8.20 (20)
Thus directions du1
dt , du2
dt and (a1, a2) are orthogonal and conjugate, i.e.
they are principal directions. Thus γ(t) is a principal curve. In the opposite
direction, if γ(t) is a principal curve then the vector dγ
dt will be conjugated
with the perpendicular vector, i.e. the second equation in (20) holds. Then
the first equation (20) implies the vector
dnγ
dt is colinear with the vector dγ
dt
for all t.
8.14 8.14 Example. Consider a surface of revolution S given by rotation JS: missing picture
of a planar curve C around an axis in the same plane and does not intersect
the curve. Similarly as in the earth, circles of latitude on S are formed
by rotation of a particular point of the curve C, wheres meridians on JS: missing picture
S are positions of the curve C in particular moments of the rotation. We
shall show that circles of latitude and meridians are principal curves of the
surface of revolution S. Consider an arbitrary meridian of the surface S
which we identify with the curve C. Thus normal vectors nC(t) of the
plane curve C are in the same time normal vectors of the surface. All vectors
nC(t) are unit thus nC(t), nC(t) = 1 shows, by differentiation, that
vectors dnC (t)
dt are perpendicular to nC(t). Thus vectors dnC (t)
dt are colinear
with tangent vectors of the curve C. Thus meridians are principal curves
according to the theorem 8.13. Circles of latitude are perpendicualr to
meridians hence theu are also principal curves (because the net of principal
curves is orthogonal).
8.15 8.15. One can derive the previous result also by computation. We prescribe
the curve C in the plane (x, z) locally by the parametrization x = g(t),
56
z = h(t), t ∈ I, i.e. the two dimensional vector g (t), h (t) is nonzero for JS: missing picture
each t ∈ I. We can assume values of the parameter t are positive. We shall
rotate along the z-axis and the assumption that C does not intersect the
axis of rotation, means that C lies in the half-plane x > 0, i.e. g(t) > 0 for
all t ∈ I. We denote by v the angle between the projection of the rotation
plane to the (x, y)-plane with the positive x-half-axis. The parameter space
D can be viewed as the space between two circles in R2 which, in polar
coordinates, is characterized by the length of the radius in the intervalu I
with arbitrary polar angle. In this sense, we can write v ∈ [0, 2π).
The point with the x-coordinate g(t) moves along the circle x = g(t) cos v,
y = g(t) sin v in the plane z = h(t). Thus the parametric description of the
surface of revolution is
f(t, v) = g(t) cos v, g(t) sin v, h(t) , t ∈ I, v ∈ [0, 2π) .
From the point if view of the general theory, t and v play the role of
parameters u1 and u2, respectively.
Partial derivatives with respect to t and v are
f1 = (g cos v, g sin v, h ) , f2 = g(− sin v, cos v, 0) .
Coefficients of the first fundamental form hence are
g11 = g 2
+ h 2
, g12 = 0 , g22 = g2
.
Further we have
f1×f2 = g(−h cos v, −h sin v, g ) , n =
1
g 2 + h 2
(−h cos v, −h sin v, g ) .
In the second order we have partial derivatives
f11 = (g cos v, g sin v, h )
f12 = g (− sin v, cos v, 0) ,
f22 = g(− cos v, − sin v, 0) .
Following 7. (5), coefficients of the second fundamenta form are
h11 =
g h − h g
g 2 + h 2
, h12 = 0 , h22 =
h g
g 2 + h 2
.
57
Generally, already conditions g12 = 0, h12 = 0 simplify the differential
equation of principal curves (9) to the form
g11 g12
h11 h22
du1 du2 = 0 .
The determinant is zero if and only if h11 = cg11, h22 = cg22, i.e. this is a
spherical or planar point which is excluded from consideration leading to
spherical curves. The equation du1du2 = 0 then characterizes the parametric
net u1 = konst. and u2 = konst. In our case of the surface of revoution,
these are meridians and circles of revolution.
8.16 8.16. We shall describe a relation between the curvature of an arbitrary
plane section of the surface S and the curvature of the normal section in
the same direction. Let be an arbitrary plane through the point p ∈ S
different form the tangent plane τpS.
Theorem (Meusnierova). Let κn be the normal curvature of the surface
S in the direction of the line ∩ τpS and 0 ≤ α < π
2 be the angle between
the normal line NpS and the plane . Then the curvature κ of the section
of the surface S by the plane at the point p satisfies
κn = κ cos α .
Proof. Let γ(s) be the arc-length parametrization of the curve ∩S, γ(0) =
p. Following the theorem 7.5, we have
κn = n,
d2γ(0)
ds2
.
It follows from the theory of planar curves that d2γ(0)
ds2 = κ e2 where e2 is a
unit vector in the plane . in our case, we have |(n, e2)| = cos α.
8.17 8.17. Consider a non-asymptotic direction A in the tangent plane. Denote
by cn the center of curvature of the normal section and by c the center
of curvature of the sectionby the plane , see the picture which shows JS: missing picture
the section by the plane perpendicular to teh direction A. It follows form
the Meusnier therem that cos α = κ
κ , hence the triangle p cn c has the
right angle at the vertex c . Geometrically this means that centers of the
curvature of all plane sections of the surface S in the direction A lie on a
circle for which the segment pcn is the diameter. Therefore given a fixed A,
the normal section has the smalles curvature and curvatures of all remaining
plane sections increases in the way described in the Meusnier theorem. JS: missing picture
An example if the sphere where these sections are circles with the radius
which decreases in this way.
58
8.18 8.18. Finally we consider a a class of surfaces which is interested both
geometrically and from a point if view of applications.
Definition. The surface S is called minimal if its mean curvature H is
zero in all points.
A nontrivial example of a minimal surface is the helicoid which we
shall study in sections 10.7 and 10.9.
The adjective “minimal” is based on the variational calculus. One of
important variational problems is the problem to “span” a surface with a
minimal are to the given curve in E3. Under rather general conditions,
solutions of this problem are surfaces with minimal mean curvature.
59
9 Envelope of a family of surfaces
60
10 Ruled surfaces
10.1. One-parameter system of lines in E3 is a mapping which each
t ∈ I maps to a line p(t) where I is an open interval. The line p(t) is called
generating line of the surface. This line can be described using a point
g(t) ∈ p(t) and a nonzero vector h(t) in the direction of p(t). An arbitrary
point of the line p(t) then has the form
e10.1e10.1 (1) f(t, v) = g(t) + vh(t) , v ∈ R .
Similarly as in agreement 1.15 or 4.7, we shall further assume g(t) and h(t)
are functions of the class Cr where r is big enough for our consideration.
Then f : I × R → E3 is map of the class Cr. Thus, in a sense, this is a
two parametric movement. The condition from the definition of surfaces
requires, beside injectivity of f, also vectors ∂f
∂t := ft = g + vh and ∂f
∂v :=
fv = h to be lienarly independent at every point. We shall illustrate this
on examples.
10.2 10.2. Let the point g(t) = a be fixed. Then we have generalized cone
e10.2e10.2 (2) f(t, v) = a + vh(t) .
Therefore ft = vh , fv = h, ft×fv = v(h ×h). Tha value v = 0 corersponds JS: missing picture
to the vertex of the cone which is obviously a singular point. Given v = 0,
we need h ×h = o for all t ∈ I. If this is satisfied then, assuming injectivity
of f, we get a surface. If h × h = 0 everuwhere, we have
e10.3e10.3 (3)
∂h
∂t
= k(t) h(t)
where k(t) is a reak function. If we consider a real function z(t) instead of
z(t) then our differential equation, by separation of variables, has the general
solution z = l(t)c where l(t) = eSk(t) dt. This holds for each component
of our vector valued function h(t) hence h(t) = l(t)b where b is a constant
vector. That is, this is the case of a two parametric movement along a line
a not a surface.
10.3 10.3. Let h(t) = a is a fixed nonzero vector. Then we get generalized
cylinder JS: missing picture
e10.4e10.4 (4) f(t, v) = g(t) + va , t ∈ I, v ∈ R .
Here ft = g , fv = a. If g (t) × a = o for all t and f is injective, it is a
surface. If the tangent vector g (t) is colinear with the vector a at some
point, this point is singular.
61
10.4 10.4. Consider a curve C ≡ g(t) given parametrically and consider the
tangent line at every point g(t). This one parametric family of curves is
called tangent developable of the curve C. Its parametric expression has
the form
e10.5e10.5 (5) f(t, v) = g(t) + v g (t) .
We have ft = g (t) + v g (t), fv = g (t) hence
e10.6e10.6 (6) ft × fv = −v g (t) × g (t)) .
Further we assume that C does not hav inflection points Then the
vector (6) is zero if and only if v = 0 which is the original point on the
curve.
Consider the normal plane ν(t, v) at the point g(t0) of the curve C.
Using scalar product, this is given by
e10.7e10.7 (7) g (t0), w − g(t0) = 0 , w(x, y, z) ∈ E3 .
The tangent line at g(t) is given parametrically as
g(t) + v g (t) .
Denote by v(t) the parameter of its intersection with ν(t0). That is,
e10.8e10.8 (8) g (t0), g(t) + v(t) g (t) − g(t0) = 0 .
Such intersection is a movement in the normal plane ν(t0, 0) given by the
parametrization
e10.9e10.9 (9) h(t) = g(t) + v(t)g (t) , v(t0) = 0 .
We shall sho that h (t0) = o. We have
h (t0) = g (t0) + v (t0) g (t0) + v(t0) g (t0) .
Since v(t0) = 0, it is enough to prove that v (t0) = −1. By differentation
of (8) we get
g (t0), g (t) + v (t) g (t) + v(t) g (t) = 0 .
Putting t = t0, we obtain
g (t0), g (t0) + v (t0) g (t0), g (t0) = 0 .
62
Since g (t0), g (t0) = 0, we have v (t0) = −1.
Considering the movement h(t), the condition h (t0) = o means that
h(t0) is a singular point. Generally, it is an edge, cf. 1.9. It is easier to think
about the tangent developable of the screw line whose projection in the JS: missing picture
direction of the axes of the screw line is on the picture. This geometrically
shows that the tangent developable is not a curve in the sense od the
definition 4.6 in a neighbourhood of the generating curve.
10.5 10.5. If one wants to obey the definition 4.6 also in the case of oneparameter
systems of lines, the following approach should be used:
Definition. A surface S ⊂ E3 is called ruled surface if S is a part of a
one-parameter system of lines.
We shall talk about a generating line of the surface S in such case
although it can be only a part of a line.
10.6. We shall further discuss two examples. First consider 3 skew lines
q1, q2, q3 (i.e. 3 lines in general position). We consider the transversal
of skew lines q1, q2 through every point a ∈ q3 (this is the intersection
of planes given by the point p and either the line q1 or q2). It is shown
in theory of the projective geometry that this construction gives rise to a
regular ruled quadric. Starting with 3 lines of the system of lines we have
just constructed and repeating this construction, we obtain the second JS: missing picture
one-parameter system of lines on the same regular ruled quadric.
10.7 10.7. Given a ruled surface which is neither a regular ruled quadric nor
a part of a plane, we have shown there can be, beside generating lines, at
most two more lines. This is related to the followng general construction.
Chose two skew lines q1, q2 and a curve C. We consider the transversal of JS: missing picture
skew lines q1, q2 through every point a ∈ C. This yields a one-parameter
system of lines
An important example (for technical applications) is the case when one
of skew lines is an improper line of some plane . Thus we have the plane JS: missing picture
, a line q and a curve C. We consider lines intersecting q and C which
are parallel with . This yields a one-parameter system of lines which is
called conoid. If the line q is perpendicular to the plane , we obtain right
conoid.
Example. IF C is the screw line, q its axis and a plane perpendicular to
q, the we obtain so called right screw conoid or helikoid. We mentioned JS: missing picture
63
this surface in 8.18. Consider q to be z-axis and C to lie on the the cylinder
with he unit radius around the z-axis. Parametric description of C then is
(cos t, sin t, bt), b = 0, t ∈ R. The helicoid has the form
e10.10e10.10 (10) f(t, v) = (v cos t, v sin t, bt) , b = 0, v ∈ R .
It is easy to see that kinematically this surface is built by screwing a
generating line perpendicular to and intersecting the axis q, in the direction
of this axis.
de10.8 10.8 Definition. A ruled surface is called developable if tangent planes
at all ponits of a fixed generating line are the same.
We say the tangent plane is tangent along generating lines.
Geometrically it is clear (and the related computation is easy) that this
is a property of generalized cylinders and cones. We shall show that also
in also in the case of tangent developable of the curve C, the tangent line
of the surface is the same at all point of a generating line. Following 10.4,
the tangent plane of the surface g(t) + vg (t) at the point g(t0) + v0g (t0),
v0 = 0 is determined by this point and vectors g (t0) and g (t0). For a fixed
t0 and each v0 = 0, this plane coincides with the osculating plane of the
curve C at the point g(t0) + vg(t0).
On the other hand, the tangent plane along a generating line of a regular
quadric is not fixed. Here the tangent plane is determined by the given
generating line and by a line of the other system through a given point.
However, the other system is formed by trasnversal lines which cannot lie
in a fixed plane. Thus considering the helicoid from 10.7, the tangent plane
changes along a generating line. It follows from (10) that
e10.11e10.11 (11) ft = (−v sin t, v cos t, b) , fv = (cos t, sin t, 0)
thus the unit normal vector is
ft × fv
ft × fv
=
1
√
v2 + b2
(−b sin t, b cos t, −v)
which changes depending on v (for a fixed t = t0).
10.9 10.9. Further we shall need to express coefficients of the second fundamental
form using the exteriorproduct We know the exterior product
[a, b, c] of three vectors in the euclidean oriented three-dimensional space
is formed id equal to the scalar product of the vector product of first vectors
with the third vector, i.e.
[a, b, c] = (a × b, c) .
64
Using this to formulae 7.(1) and 7.(5), we obtain
e10.12e10.12 (12)
h11 =
1
f1 × f2
[f1, f2, f11] ,
h12 =
1
f1 × f2
[f1, f2, f12] ,
h22 =
1
f1 × f2
[f1, f2, f22] .
Example. We shall show the helicoid from (10) is a minimal surface in
the sense of 8.18, i.e. that H = 0. Differentiating of (11) we obtain ftt =
(−v cos t, −v sin t, 0), ftv = (− sin t, cos t, 0) and fvv = o. Thus h11 = 0 and
h22 = 0. Since also g12 = 0, the formula 8.(15) yields H = 0.
10.10 10.10. Since every line on a surface is asymptotic curve, all points on ruled
surfaces are hyperbolic, parabolic or planar.
Theorem. A developable ruled surfaces S has zero Gaussian curvature.
Proof. Given a parametrization f(t, v) = g(t) + v h(t) of the surface plochy
S, we have f1 = g (t) + v h (t), f2 = h(t). For a fixed t, the vector h(t)
lies in the associated vector space of the tangent plane for all v if for the
vectors h(t) and g (t)+v1h (t), g (t)+v2h (t) are complanar for all v1 = v2.
By a lienar combination of these vectors we obtain g (t) and h (t) hence
the condition for a fixed tangent plane is
e10.13e10.13 (13) g (t), h(t), h (t) = 0 .
A further computation yiedls f11 = g (t) + v h (t) and f12 = h (t),
f22 = o. Following (12) we obtain h22 = 0,
h12 =
1
f1 × f2
g (t) + v h (t), h(t), h (t)
and also h12 = 0 according to (13). It follows from 8.(15) that K = 0
independently on h11.
10.11 10.11. In the opposite direction we have the following statement.
Theorem. If every point of the surface S is parabolic then S is locally a
developable ruled surface.
65
Proof. Given a parabolic surface S, we chose local paramatrization such
that the net of asymptotic curves is given by u1 = konst.. Then 0 = h12 =
(n, f12) and 0 = h22 = (n, f22). We know that (n, f1) = 0, (n, f2) = 0,
(n, n) = 1. Differentiating this with respect to u2 yields – similarly as in
7.(8) – that
(n2, f1) = 0 , (n2, f2) = 0 , (n2, n) = 0 .
Thus n2 = o, i.e.. the normal vector along the curve u1 = konst. is fixed.
Followign Podle 7.(8), it follows from h12 = 0 that (n1, f2)=0. Differenting
this with respect to u2 and using u12 = o, we obtain (n1, f22 = 0). Hence
vectors f2 and f22 are perpendicular to vectors n and n1 which are linearly
independent. Indeed, the vector n1 is perpendicular to n and is nonzero.
Indeed, the first of equations in 7.(8) means (n1, f1) + (n, f11) = 0. Thus
n1 = o would mean h11 = 0 which, together with h12 = 0 and h22 = 0,
would identify a planar point which we exclude. Sicne vectors f2 and f22
are perpendicular to two linearly independent vectors, they are colinear.
Hence every point of the curve u1 = konst. is inflection, i.e. this curve is
a line. The tangent line along this generating line is constant thus S is
locally a developable ruled surface.
de10.12 10.12 Definition. Generating line g(t0) + vh(t0) of a ruled surface (1)
is called cylindric if vectors h (t0) and h(t0) are colinear.
Theorem. A ruled surface with all generating lines cylindric is a generalized
cylinder.
Proof. We have shown in 10.2 that it follows from dh
dt = k(t) h(t) that
h(t) = l(t)b where b is a constant vector. Thus we have
f(t, v) = g(t) + v l(t)b
which is a cylinder with a different parametrization of generating lines.
10.13 10.13. Similarly, we have a direct geometrical characterization of a cylindrical
line. Given a parametrization p(t) ≡ g(t) + v h(t), we can assume
that the vector h(t) is unit. Then h(t) is a motion along a unit sphere
which is called spherical image of a ruled surface.
Differentiating (h, h) = 1 we obtain (h, h ) = 0 thus the vector h (t) is
perpendicular to h(t) for eveery t. Assuming further that h (t0) is colinear
with h(t0), we obtain h (t0) = o. Therefore
66
Theorem. Generating line p(t0) of a ruled surface S is cylindric if and
only if t0 is a singular point of the sphercial image of the surface S.
10.14 10.14. The meaning of the word “generally” in the following statement is
explained during the proof.
Theorem. A ruled surface without cylindrical lines is generally either a
tangent developable or a cone.
Proof.
10.15 10.15. Our results from 10.12 and 10.14 are sometimes summarized as
surface with zero Gaussian curvature is generally either a tangent
developable or a generalized cylinder or a generalized cone.
67
11 Isometric mappings
11.1 11.1. Let D, ¯D ⊂ R2 be open sets. A map ϕ: D → ¯D is given by a pair
of functions ϕ1, ϕ2 : D → R which are called components of the map ϕ.
Denoting u1, u2 coordinates in D and v1, v2 coordinates in ¯D, we have
v1 = ϕ1(u1, u2), v2 = ϕ2(u1, u2).
Definition. We say ϕ is mapping of the class Cr if ϕ1 and ϕ2 are
functions of the class Cr.
Henceforth we assume ϕ is a mapping of the class Cr, r ≥ 1.
de11.2 11.2 Definition. The determinant J(ϕ) =
∂ϕ1
∂u1
, ∂ϕ1
∂u2
∂ϕ2
∂u1
, ∂ϕ2
∂u2
is called Jacobian
of the mapping ϕ.
11.3 11.3. We shall study mapppings g: S → ¯S between two simple surfaces of
the class Cr. Assume S and ¯S are given by parametrizations f(u1, u2), JS: missing picture
(u1, u2) ∈ D and ¯f(v1, v2), (v1, v2) ∈ ¯D. The map g determines a unique
map ψ: D → ¯D such that g◦f = ¯f ◦ψ. Thus ψ expresses g in the parameter
space. In the opposite direction, ψ deteremines the mapping g.
Definition. We say that g: S → ¯S is mapping of the class Cr if the
corresponding mapping ψ: D → ¯D is of the class Cr.
11.4 11.4. A choice of parametrizations of surfaces S and ¯S is used in the
previous definition. We shall show that the class of differentiability of
the mapping g is independent on the choice of parametrizations f and ¯f.
First we discuss a change of the parametrization f(u1, u2) of the surface S,
(u1, u2) ∈ D.
Consider a bijective mapping ϕ: ¯D → D of the class Cr where ϕ =
ϕ1(v1, v2), ϕ2(v1, v2) , (v1, v2) ∈ ¯D. Then ¯f = f ◦ ϕ is also of the class Cr.
In the case of parametrziation of a surface we have further the condition
f1 × f2 = o. Using the notation ¯f1 = ∂1(f ◦ ϕ), ¯f2 = ∂2(f ◦ ϕ) and using
the chain rule, we obtain
e11.1e11.1 (1) ¯f1 = f1
∂ϕ1
∂v1
+ f2
∂ϕ2
∂v1
, ¯f2 = f1
∂ϕ1
∂v2
+ f2
∂ϕ2
∂v2
.
Thus
¯f1 × ¯f2 =
∂ϕ1
∂v1
∂ϕ2
∂v2
−
∂ϕ2
∂v1
∂ϕ1
∂v2
f1 × f2 = J(ϕ) f1 × f2 .
68
Definition. A bijective map ϕ: ¯D → D of the class Cr is called reparametrization
if J(ϕ) = 0for all (v1, v2) ∈ ¯D.
11.5 11.5. Now it is clear that the definition 11.3 does not depend on the choice
of parametrization. Consider a reparametrization ϕ: D1 → D on S and a
reparametrizaci ¯ϕ: ¯D1 → ¯D on ¯S. Then ¯ϕ−1 : ¯D → ¯D1 is also a map of the
class Cr (this follows immediately from the generalized implicit function
theorem, cf. e.g. the textbook [5]). Thus we have ¯ϕ−1 ◦ ψ ◦ ϕ instead of the
mapping ψ in the definition 11.3 which is also of the class Cr.
11.6 11.6. Further we shall consider motion on surfaces.
First consider the map ψ: D → ¯D, v1 = ψ1(u1, u2), v2 = ψ2(u1, u2).
Then D itself is a surface (as a part of plane) hence we have the tangent
plane TuD for every u ∈ D which coincides with R2. Its elements are
tangent vectors to motions h(t), h: I → D at the point h(0) = u where we
assume 0 ∈ I. Coordinates of the tangent vector are dh1(0)
dt , dh2(0)
dt . Consider
the motion ψ ◦h: I → ¯D. Coordinates of its tangent vector at t = 0, which
one findss applying the chain rule to ψ1 h1(t)h2(t) and ψ2 h1(t), h2(t) ,
are
e11.2e11.2 (2)
∂ψ1
∂u1
dh1(0)
dt
+
∂ψ1
∂u2
dh2(0)
dt
,
∂ψ2
∂u1
dh1(0)
dt
+
∂ψ2
∂u2
dh2(0)
dt
.
That is, the tangent vector d(ψ◦h)(0)
dt is determined by the vector dh(0)
dt .
Considering the linearity of (2), we have
Theorem. The rule dh(0)
dt → d(ψ◦h)(0)
dt deteremines a linear mapping Tuψ: TuD →
Tψ(u)
¯D for each u ∈ D.
Definition. This mapping is called tangent mappping ψ at the point u.
Denoting by (du1, du2) coordinates at TuD by (dv1, dv2) coordinates in
Tψ(u)
¯D then (2) has the form
e11.3e11.3 (3) dv1 =
∂ψ1
∂u1
du1 +
∂ψ1
∂u2
du2 , dv2 =
∂ψ2
∂u1
du1 +
∂ψ2
∂u2
du2 .
Thus these are differentials of functions ψ1 and ψ2.
11.7 11.7. Consider the original map g: S → ¯S and put p = f(u) ∈ S. Consider
the tangent space TpS and a vector vektor A ∈ TpS which is tangent to the
motion γ(t) on S, A = dγ(0)
dt . Then g ◦ γ is a motion on ¯S and the tangent
vector d(g◦γ)(0)
dt to this motion depends only on A. Indeed, it is given (2) in
terms of the parametrization. Thus we have shown
69
Theorem. The rule dγ(0)
dt → d(g◦γ)(0)
dt determines a linear map Tpg: TpS →
Tg(p)
¯S.
Definition. The map Tpg is called tangent map to g at the point p.
de11.8 11.8 Definition. We say the mapping g: S → ¯S is isometric if each
tangent map Tpg: TpS → Tp
¯S, p ∈ S preserves the scalar product.
That is, (A, B) = Tpg(A), Tpg(B) for each A, B ∈ TpS.
If g is bijective, it is isometry of S and ¯S.
11.9 11.9. The bijiective map g: S → ¯S can be realised in sucha way we cos- JS: missing picture
nider a common parameter space D and corresponding points f(u1, u2) ∈ S
and ¯f(u1, u2) ∈ ¯S. In this case we say the map g is given by equality of
parameters.
Theorem. The bijection g: S → ¯S given by equality of parameters is
isometry if and only if first fundamental forms Φ1 and ¯Φ1 of surfaces S and
¯S, respectively are coincide.
Proof. Vectors f1, f2 form a basis in TpS, vectors ¯f1, ¯f2 form a basis in
Tg(p)
¯S and the linear map Tpg is given by du1 = du1, du2 = du2. Following
6.1, scalar products of tangent vectors to S and ¯S are given by the first
fundamental form.
The condition Φ1 = ¯Φ1 explicitly means g11 = ¯g11, g12 = ¯g12, g22 = ¯g22
where bared quantities are computed on the surface ¯S depending on the
same parameters (u1, u2).
11.10 11.10. The following statement justifies the terminology of isometry.
Theorem. The bijection g: S → ¯S is isometry if and only if it preserves
length of curves.
Proof. The bijection g can be given by equality of parameters. Since the
length of the curve u1(t), u2(t) , t ∈ [a, b] is
e11.4e11.4 (4) s =
b
a
g11
du1
dt
2
+ 2g12
du1
dt
du2
dt
+ g22
du2
dt
2
dt ,
it follows from the theorem 11.9 that length of curves corresponding to
ismoetries are the same. In the opposite direction, if both curves with the
same parametric expression have the same length, also their tangent vectors
have the same length according to (4). Thus the linear map Tpg for each
p ∈ S preserves length of vectors. Now it follows from linear algebra theory
that such mappresrves also the scalar product.
70
11.11 11.11. Geometrically it is obvious that the cylinder f(u) = (r cos u1, r sin u1, u2),
u1 ∈ (0, 2π), u2 ∈ R is isometric with a plane strip with the same coordinates.
Physically, this isometry is given by unfolding the cylinder into
the plane. We verify this also by computation using the theorem 11.9. We
have f1 = (−r sin u1, r cos u1, 0), f2 = (0, 0, 1) hence g11 = r2, g12 = 0,
g22 = 1. The plane z = 0 can be parametrized as ¯f(u) = (ru1, u2, 0). Thus
¯f1 = (r, 0, 0), ¯f2 = (0, 1, 0) and ¯g11 = r2, ¯g12 = 0, ¯g22 = 1 which is the same
as in the case of cylinder.
de11.12 11.12 Definition. By inner geometry of the surface S we mean properties
preserved by isometries.
It follows from the theorem 11.9 that inner geometry of the surface
is formed by properties which can be derived from the first fundamental
form. We say that properties of S which essentially depend on the second
fundamental form, belong to outer geometry of the surface.
11.13 11.13. The notion of inner geometry of the surface originates in Gauss’
work. He derived the following its most important statement. It is a deep
result which will be proved in the next section in 12.14. Gauss called this
Teorema egregium in latin (the transaltion: remarkable theorem). Let KS
be the Gaussian curvature of the surface S K¯S the Gaussian curvature of
the surface ¯S.
Teorema egregium (Gauss). If g: S → ¯S is an isometry then KS(p) =
K¯S g(p) for all p ∈ S.
One should emphasize that both principal curvatures do not belong to
inner geometry of the surface (the second fundamental form is essential for
their computatin) whereas their product does.
11.14 11.14 Example. An open set in the plane cannot be isometric to an open
set on the sphere. Indeed, The Gaussian curvature of the plane is zero
whereas the Gaussian curvature of the sphere of the radius r is 1
r2 .
11.15 11.15. Recall by neighbourhood on the surface S of the point p ∈ S we
mean intersection of a neighbourhood of p in E3 with the surface S.
Definition. The surface S is called developable if each point p ∈ S has
a neighbourhood isometric to an open set in the plane.
This isometry is understood as “unfoldability” of the corresponding
part of the surface into the plane, see the example 11.11 which justifies this
terminology. On the other hand, we introduced the notion of a developable
71
ruled surface at 10.8. The same terminology is based on the fact that
both notions coincide as we shall show. If necessary to distinguish both
definitions, the latter one will be referred as developability in a metric
sense.
11.16 11.16. Consider a generalized cylinder. Assume the z-axis is parallel with
generating lines and the curve g is the section of the cylinder by the
plane z = 0 parametrized by the arc-length. The the cylinder is given
by parametrization
f(s, v) = g1(s), g2(s), v .
Thus f1 = (g1, g2, 0), f2 = (0, 0, 1), g11 = 1, g12 = 0, g22 = 1. Using
the parametrization ¯f(s, v) = (s, v, 0) of the plane, we obtain the same
coefficients ¯g11 = 1, ¯g12 = 0, ¯g22 = 1. Hence equality of parameters yields
an isometry of both surfaces.
11.17 11.17. Consider a generalized cone with the vertex at the origin. Thus
f(t, v) = g(t) v. Here we can assume the curve g is parametrized by arclength
s and g(s) = 1 for all s. Then f1 = g (s) v, f2 = g(s), i.e. g11 = v2,
g22 = 1 and g12 = 0 since vectors g(s) and g (s) are perpendicular. If we
choose g as the unit circle k(s) in the plane we can parametrized the plane JS: missing picture
locally in the form ¯f(s, v) = k(s) v. Hence ¯g11 = v2, ¯g12 = 0, ¯g22 = 1 which
shows a local isometry of the plane and the cone.
11.18 11.18. Consider the tangent developable g(t) + vg (t) of the curve C. We
assume C is parametrized by arc-length. Using Frenet formuale, we can
parametrize the surface as
f(s, v) = g(s) + ve1(s) .
Thus f1 = e1(s) + vκ(s) e2(s), f2 = e1(s), i.e. g11 = 1 + v2 κ2(s), g12 = 1,
g22 = 1. Further consider a curve ¯g(s) in the plane which has locally the
same curvature as the spacial curve C and parametrize the plane locally as
¯f(s, v) = ¯g(s) + v ¯g (s). This can be written also in the form
¯f(s, v) = ¯g(s) + v ¯e1(s) .
We have ¯f1 = ¯e1(s) + v κ(s) ¯e2(s), ¯f2 = ¯e1(s). Also in this case we have
¯g11 = 1 + v2κ2(s), ¯g12 = 1, ¯g22 = 1. Thus we obtained a local isometry of
teh tangent developable with the plane.
11.19 11.19. We have shown in 10.11 that a surface with zero Gaussian curvature
without planar points is locally a developable ruled surface. We know
72
from 10.15 that a developable ruled surface is generally either tangent developable
orgeneralized cylinder or generalized conev. We have shown these
surfaces are developable in the metric sense. Using an alternative approach
(which we shall not discuss here) can be shown the following statement (cf.
the theorem 14.6).
Theorem. A surface S is locally isometric to the plane if and only if it has
zero Gaussian curvature.
73
12 Parallel transport of vectors on a surface
12transport
12.1 12.1. Consider a surface S ⊂ E3 and a motion γ : I → S. We denote by V
the associated vector space of E3.
Definition. The mapping A: I → V is callede tangent vector field on
S aling the motion γ if A(t) ∈ Tγ(t)S for all t ∈ I.
Zero tangent vectors along γ form the field denoted by 0γ. If A is a
tangent vector field along γ and g: I → R is a function then g(t)A(t) is
also a tangent vector field along γ. If A1 and A2 are two tangent vector
fields along γ then also A1(t) + A2(t) is a tangent vector field along γ.
12.2 12.2. Recall NpS is the normal vector space of the surface S at the point p.
The following definition is essential for the differential geometry of surfaces.
Definition. We say that tangent vectior field A along the motion γ
is formed by vectors parallel on S if dA
dt ∈ Nγ(t)S for all t ∈ I.
12.3 12.3. The vector dA
dt decomposes into a direction in the tangent plane Tγ(t)S
and the normal line Nγ(t)S at each point of the motion γ(t). TTangent
components form again a tangent vector field on S along γ.
Definition. Tangent vectir field A
dt on S along γ formed by tangent components
of vectors dA
dt is called textbfcovariant derivative of the tangent
vector field A on S along the motion γ.
Thus the field A is formed by vectors parallel on S if and only if A
dt is
the zero field along γ.
12.4 12.4. We shall find a parametric expression for A
dt . Since vectors f1, f2,
n are linearly independent at each point of the surface, we can decompose
second partial derivatives as
e12.1e12.1 (1)
f11 = Γ1
11(u1, u2) f1 + Γ2
11(u1, u2) f2 + h11(u1, u2) n ,
f12 = Γ1
12(u1, u2) f1 + Γ2
12(u1, u2) f2 + h12(u1, u2) n ,
f22 = Γ1
22(u1, u2) f1 + Γ2
22(u1, u2) f2 + h22(u1, u2) n .
Taking scalar product of each equation with the unit vector n perpendicular
to f1 and f2 shows that coefficients at n are indeed coefficients of the second
fundamental form as the notation indicates.
74
Definition. The function Γi
jk, i, j, k = 1, 2, Γi
21 = Γi
12 are called Christoffel
symbols of the surface S corresponding to the parametrization f(u1, u2).
12.5 12.5. Let the motion γ(t) is given by the parametrization u1(t), u2(t)
and A(t) = U1(t), U2(t) . Then
e12.2e12.2 (2) A(t) = U1(t) f1 u1(t), u2(t) + U2(t) f2 u1(t), u2(t) .
From this one directly computes using (1) that
dA
dt
=
dU1
dt
f1 + U1 f11
du1
dt
+ f12
du2
dt
+
dU2
dt
f2 + U2 f12
du1
dt
+ f22
du2
dt
=
dU1
dt
+ Γ1
11U1
du1
dt
+ Γ1
12 U1
du2
dt
+ U2
du1
dt
+ Γ1
22U2
du2
dt
f1
+
dU2
dt
+ Γ2
11U1
du1
dt
+ Γ2
12 U1
du2
dt
+ U2
du1
dt
+ Γ2
22U2
du2
dt
f2 + (. . . )n .
where an expression at n is not important for us. Denoting U1
dt , U2
dt
coordinates of the vector field A
dt along γ, we have
e12.3e12.3 (3)
U1
dt
=
dU1
dt
+
2
i,j=1
Γ1
ij u(t) Ui
duj
dt
,
U2
dt
=
dU2
dt
+
2
i,j=1
Γ2
ij u(t) Ui
duj
dt
.
12.6 12.6. As the first property of formulae (3) we shal derive th following:
Theorem. Tangent vector fields A, B on S along the motion γ and the
function g: I → R satisfy
e12.5e12.5 (4)
(A + B)
dt
=
A
dt
+
B
dt
,
(gA)
dt
=
dg
dt
A + g
A
dt
.
Proof. This follows directly from (3).
12.7 12.7. The following statement shows that the parallel transport along a
given motion has similar properties as the parallel transport of vectors in
the plane.
Theorem. For each motion γ : I → S, each t0 ∈ I and each vector
A0 ∈ Tγ(t0)S exists unique vectir field along γ satisfying A(t0) = A0 and
formed by vectors parallel on S.
75
Proof. Given the motion γ, the condition that right hand sides of expressions
of (3) foms a system of two ordinary differential equations. The value
A0 is the initial condition for this system. This determines the solution
uniquelly.
de12.8 12.8 Definition. We say the tangent vector field A from the theorem 12.7
provides parallel transport of the vector A along the motion γ on
S.
Assume the motion γ(t) is the parametrized simple curve C on S. A
reparametrization t = ϕ(τ) of C yields another motion γ ◦ ϕ. Inserting
this reparametrization into (3), derivation with respect to t are mutliplied
by dϕ
dτ . Since expressions (3) are linear in dUi
dt a dui
dt , i = 1, 2, differential
equations Ui
dt = 0, i = 1, 2 for parallel transport are essentially the same
as (Ui◦ϕ)
dτ = 0. Geometrically this means for different parametrizations of
the curve C on S we get the same parallel transport of vectors. Thus we
speak not only about paralle transport of vectors along a motion on S but
also about parallel transport of vectors along a curve on S.
12.9 12.9 Theorem. If tangent vecros fields A and B along the motion γ provide
parallel transport of vectors A0 ∈ Tγ(t0)S and B0 ∈ Tγ(t0)S, respectively,
then the field k1A +k2B provides the parallel transport of the vector
k1A0 + k2B0, k1, k2 ∈ R.
Proof. The theorem 12.6 for constant g = k yields (kA)
dt = k A
dt . Thus
(k1A+k2B)
dt = k1
A
dt + k2
B
dt . Since the right hand side is zero, the same
holds for the left hand side.
Geometrically this means the parallel transport preserves lienar combinations
of vectors.
12.10 12.10 Example. Considering a plane as a surface in E3 then for the
tangent vector field A(t) = U1(t), U2(t) along an arbitrary motion γ(t)
in has zero normal component of the vector dA
dt . Thus A(t) is parallel
transport along γ if and only if dA
dt = 0, i.e. U1(t) a U2(t) are constants.
This is the classical parallel transport in the plane. This transport does
not depend on the motion.
Now we shall show that, however, parallel transport on the sphere S
does depend on the motion. Conside one eighth of the sphere according
to the picture. The great circle in the plane z = 0 has the paramet- JS: missing picture
ric expression f(t) = (r cos t, r sin t). Its tangent vector is v(t) = df
dt =
76
(−r sin t, r cos t). Thus dv
dt = (−r cos t, −r sin t). The normal vector of the
sphere at the point f(t) is (cos t, sin t) hence dv
dt ∈ Nf(t)S. Therefore tangent
vectors to the great circle provide transport parallely along this circle.
Denote by a the point with the parameter t = 0 and b the point with the
parameter t = π
2 .
Now let us transport this tangent vector v = v(0) along the great circle
in the plane perpendicular to v at the point c with the parameter π
2 . The
constant vector v is tangent along this curve, i.e. dv
dt = o. Now let us
continue with the parallel transport along the small arc of the great circle
from the point c to b. Here we again transport the tangent vector of the
circle along this circle hence the result is again tangent to this circle. Thus
the resulting transport of the vector v from the point a to b along two
different motions I and II give rise to two different vectors which are actualy
perpendicular. Although the second motion is not differentiable at one
point, the “vertex” at the point c is not the reason of different parallel
transports of the vector v along motions I and II.
12.11 12.11. Decompositions (12.1) can be used also to compute Christoffel symbols.
This is particularly simple in the case when the parametric net is
orthogonal, i.e. g12 = (f1, f2) = 0. Let us discuss the sphere f(u1, u2) =
r(cos u1 cos u2, sin u1 cos u2, sin u2) from 6.4. We found
f1 = r(− sin u1 cos u2, cos u1 cos u2, 0) ,
f2 = r(− cos u1 sin u2, − sin u1 sin u2, cos u2) ,(5)
g11 = (f1, f1) = r2
cos2
u2 , g12 = (f1, f2) = 0 , g22 = (f2, f2) = r2
.
Further differentiation yields
e12.6e12.6 (6)
f11 = r(− cos u1 cos u2, − sin u1 cos u2, 0) ,
f12 = r(sin u1 sin u2, − cos u1 sin u2, 0) ,
f22 = r(− cos u1 cos u2, − sin u1 cos u2, − sin u2) .
We consider scalar product of equations (1) with vectors f1 and f2 where
we need to compute scalar products on the left hand side. We obtain
0 = Γ1
11r2
cos2
u2, r2
sin u2 cos u2 = Γ2
11r2
, −r2
sin u0 cos u0 = Γ1
12r2
cos2
u2 ,
0 = Γ2
12r2
, 0 = Γ1
22r2
cos u2 , 0 = Γ2
22r2
.
Thus
Γ1
11 = 0 , Γ2
11 = sin u2 cos u2 , Γ1
12 = − tan u2 , Γ2
12 = 0 , Γ1
22 = 0 , Γ2
22 = 0 .
77
12.12 12.12. The following theorem is an important tehoretical result. Equations
(12.7) show that Christoffel symbols can be expressed using coefficirnts of
the first fundamental form. THUS THE PARALLEL TRANSPORT OF
VECTOS ON THE SURFACE BELONGS TO THE INNER GEOMETRY
OF THE SURFACE.
Since g11g22 −g2
12 > 0, the square matrix (2×2)-matrix (gij) is reguklar.
Denote by (gkl) its inverse matrix.
Theorem. We have
e12.7e12.7 (7) Γk
ij =
1
2
2
l=1
gkl
∂gjl
∂ui
+
∂gli
∂uj
−
∂gij
∂ul
.
Proof. Differentiating (f1, fj) = gij with respect to ul, we obtain
e12.8e12.8 (8)
∂gij
∂ul
= (fil, fj) + (fi, fjl) .
It follows from (1) that fij =
2
m=1
Γm
ij fm + hijn. By substitution we obtain
(fil, fj) =
2
m=1
Γm
il gmj .
Thus (8) can be written as
e12.9e12.9 (9)
∂gij
∂ul
=
2
m=1
Γm
il gmj + Γm
jl gmi .
We obtain two more equations by a suitable change of indices, Dal? dv?
rovnice z?sk me z m?nou index?
∂gil
∂uj
=
2
m=1
Γm
ij gml + Γm
lj gmi ,e12.10e12.10 (10)
∂glj
∂ui
=
2
m=1
Γm
ji gml + Γm
li gmj .e12.11e12.11 (11)
Summing (10) + (11) − (9) and using symmetris of gij and Γk
ij at lower
indices, we obtain
e12.12e12.12 (12)
∂gil
∂uj
+
∂glj
∂ui
−
∂gij
∂ul
= 2
2
m=1
Γm
ij gml .
78
By division by 2 and for fixed i, j, we consider (Γm
ij ) as rank two row vector.
Then the matrix form of the right hand side of (12) is (Γm
ij )(gml). Now we
compute the unkown Γm
ij by multiplication of the inverse matrix (gkl). This
yields (7).
12.13 12.13. Of course, (7) can be used also as a formula to compute Christoffel
symbols. A simple example is the generalized cylinder from 11.16. where
we found g11 = 1, g12 = 0, g22 = 1. All partial derivatives in (7) are thus
zero since these are derivatives of a constant. Hence all Christoffel symbols
of the generalized cylinder are zero.
12.14 12.14. Now we shall prove the Gauss’ theorem egregium. We shall start
with equations (1) which we shall wwrite in the form
e12.13e12.13 (13) fij =
2
l=1
Γl
ijfl + hijn .
We shall use the notation fijk = ∂3f
∂ui∂uj∂uk
, Γl
ij,k =
∂Γl
ij
∂uk
, ni = ∂n
∂ui
, hij,k =
∂hij
∂uk
. Equations 7.(8) can be written in the form
e12.14e12.14 (14) (ni, fj) = −hij .
Differentiating (13) with respect to uk, we obtain
e12.15e12.15 (15) fijk =
2
m=1
Γm
ij,kfm +
2
n=1
Γn
ijfnk + hij,kn + hijnk .
Using (13), the second term on the right hand side has the form
e12.16e12.16 (16)
2
n=1
Γn
ijfnk =
2
m,n=1
Γn
ij(Γm
nkfm + hnkn) .
Taking the scalar product of (15) with the vector fj and using (14), we
obtain
e12.17e12.17 (17) (fijk, fl) =
2
m=1
Γm
ij,kgml +
2
m,n=1
Γn
ijΓm
nkgml − hijhkl .
It follows from symmetry of third partial derivatives fikj = fijk that (17)
holds also after exchange of indices j and k. Subtracting these two equations,
we obtain
e12.18e12.18 (18) hijhkl − hikhjl =
2
m=1
gml Γm
ij,k − Γm
ik,j +
2
n=1
(Γn
ijΓm
nk − Γn
ikΓm
nj) .
79
Putting i = 1, j = 1, k = 2, l = 2 we obtain
Theorem (Gauss’ equations). It holds
e12.19e12.19 (19) h11h22 − h2
12 =
2
m=1
gm2 Γm
11,2 − Γm
12,1 +
2
n=1
(Γn
11Γm
n2 − Γn
12Γm
n1) .
The right hand side depends only on coefficients gij of the first fundamental
form and its partial derivatives of thr first and second order according
to the theorem 12.12. We have found in 8.9 that K = (h11h22 −
h2
12)/(g11g22 − g2
12). Thus the Gaussian curvature K belongs to the inner
geometry of the surface as the theorema egregium states.
We also remark that putting different indices i, j, k, l into (18), we
obtain again the equation (19) or identity.
12.15. Information If we analogously decompose ni, i = 1, 2 into the12.15
frame f1, f2, n and use the relation ∂ni
∂uj
=
∂nj
∂ui
= ∂2n
∂ui∂uj
, we obtain so
called Codazzi equations (two of them are essentially)
e12.20e12.20 (20) hij,k − hik,j +
2
l=1
(Γl
ijhlk − Γl
ikhlj) = 0 .
Using elementary techniques for systems of partial differential equations
(e.g. the theorem 7.8 in the textbook [5]), one can prove following two statements
about existence and uniqueness of solutions, which are sometimes
called The basic theorem of theory of surfaces.
I. If S and ¯S are two simple surfaces with parametrizations f : D → E3
and ¯f : D → E3, respectively on the same parameter space D, which have
the same first and second fundamental form, then there exists an Euclidean
motion ϕ: E3 → E3 such that ϕ ◦ f = ¯f.
Summarizing: surfaces, which have the same first and second fundamentzal
form, are congruent.
II. Consider two quadratic forms Φ1 and Φ2 on D ⊂ R2 where Φ1 is
positive definite at all points. If Φ1 and Φ2 satisfy Gauss and Codazzi equations,
then there locally exists a surface with a parametrization f : D → E3
such that Φ1 and Φ2 are its first and second fundamental form, respectively.
80
13 Geodetic curves
13
13.1 13.1. Given a surface S and a motion γ : I → S, we can consider the field
of tangent vectors of γ which we denote by ˙γ.
Definition. The motion γ : U → S is called geodetic curve if the field ˙γ
of its tanegnt vectors transports parallely along γ.
Consider a plane as a surface, this definition means that the vector
˙γ = a is constant, i.e. γ is the motion p + ta along a line, p ∈ .
13.2 13.2. Thus the condition for the motion γ to be geodetic means ˙γ
dt = o.
Let γ(t) = u1(t), u2(t) . Hence we need to put γ(t) = u1(t), u2(t) into
the relation .(12.3) and require the right hand side to be zero. I.e. geodetic12
motions are solutions of a system of two differential equations of the 2.
order:
e13.1e13.1 (1)
d2ui
dt2
+
2
j,k=1
Γi
jk u(t)
duj
dt
duk
dt
= 0 , i = 1, 2 .
This system behaves more or less similarly as one differential equations of
teh 2. order.
13.3 13.3. It is well known that a solution of a differential equation of the 2.
order is fully determined by its initial value and initial velocity (i.e. the
value of its derivative). Analogously, in teh case of the system (13.1) we
have
Theorem. For every point p ∈ S and every vector A ∈ TpS there exists an
interval 0 ∈ IA ⊂ R and a unique geodetic motion γA : IA → S such that
γ(0) = p and ˙γ(0) = A.
The interval IA generally changes depending on A.
13.4 13.4. It is useful to observe the following property.
Lemma. If γ(t) is a geodetic motion then also the motion γ(at + b) is
geodetic for each a, b ∈ R, a = 0.
Proof. Put γ(t) = γ(at + b). We have dγ
dt = dγ
dt a, d2γ(t)
dt2 = d2γ
dt2 a2. Multiplying
equations by a2 then also γ(t) = u1(t), u2(t) satisfiese13.1
d2ui
dt2
+
2
j,k=1
Γi
jk u(t)
duj
dt
duk
dt
= 0 , i = 1, 2 .
81
This lemma has a natural kinematic interpretation in the plane: If we
change parametrization of a linear stadey motion, we obtain again a linear
stady motion.
13.5 13.5 Definition. The curve C ⊂ S is called geodetic curve, if there
exists such a parametrization γ(t) that γ is a geodetic motion.
We briefly talk about geodetics. Geodetic curves in the plane are lines.
It follows from 13.3 and 13.4 that
Theorem. For each point p ∈ S and each direction in TpS there exists a
unique geodetic on S which touches this direction at the point p.
13.6 13.6. It follows form properties of solutions of systems of differential equations
of the 2. order (which we shall not discuss here in detail) that
Theorem. At each point p ∈ S there exists a neighbourhood Z ⊂ S such
that for each two points q1 = q2 in Z there exists a unique geodetic in Z
through points q1 and q2.
This property is analogous to the fact that each two points in the plane
can be connected by a unique line. However, a simple examples shows that
the locality assumption in the theorem is essential on surfaces. We shall
show below that geodetic curves on the sphere S are great circles. Given
any point q1 ∈ S and another point q2 (different from the “opposite pole”),
there is a unique great circe throught q1 and q2. However, if q2 is the
“opposite pole” to q1 then there are infinitely many great circles throught
points q1 and q2.
13.7 13.7. Recall the osculatung plane of a curve is not defined in inflection
points. The following theorem provides “outer” characterization of geodesics.
Theorem. The curve C ⊂ S is geodesic if and only of its osculating plane
at each point contains the normal direction of the surface or is not defined.
Proof. The condition ˙γ
dt = o means that the vector lies in the normal
direction. If d ˙γ
dt = o then vectors ˙γ and d ˙γ
dt determine the osculating plane
which contains the normal direction. If d ˙γ
dt = o then this is teh inflection
point. In the opposite direction, consider a curve C parametrizad by the
arc-length γ(s). Then ˙γ(s), ˙γ(s) = 1. Differentiating the latter, we obtain
˙γ, d ˙γ
ds = 0. Je-li d ˙γ
ds = o then vectors ˙γ and d ˙γ
ds determine the osculating
plane and we assume this plane contains the normal line of the surface.
Since the vector d ˙γ
ds is perpendicular to the vector ˙γ(s), it is parallel with
the normal line. Thus ˙γ
ds = o. If d ˙γ
ds = o then ˙γ
ds = o.
82
Example. The great circle C on the sphere S is such circle whose center
coincides with the center of the sphere. Its usculating plane coincides with
the plane the circle lies in (at every point). Normal lines of the sphere
along C lie in the same plane. Hence each great circle is a geodesic. On
the other hand, at each point p ∈ S and each direction in TpS, there is a
unique great circle. Using the theorem 13.5, other geodesics on S do not
exist.
13.8 13.8 Corollary. If C is a geodesic curve and γ(s) its arc-length parametrization
then γ(s) is a geodesic motion.
Proof. It follows from the theorem 13.7, the osculating plane of the curve
contains the normal direction of the surface. The second part of the proof
shows that ˙γ
ds = o.
13.9 13.9. Let Z ⊂ S is a neighbourhood of the point p ∈ S with the property
from the theorem 13.6
Theorem. For each q ∈ Z, q = p, the geodesic through points p and q is
the shortes curve in Z which connects points p and q.
JS: missing picture
Proof. Let us denote by C the geodesic connecting points p and q. Choose
a curve ¯C through the point p perpendicular to C. Further, we have a
geodetic curve through every point of ¯C perpendicular to ¯C; these curves
form a 1-parameter family of parametric curves. We choose its orthogonal
trajectories as the second family of parametric curves. Choosing a parametr
u1 on C and a parameter u2 on ¯C, we obtain a parametrization of the
surface on some neighbourhood U of the point p. We can put p = (0, 0),
q = (a, 0), a > 0.
Parametrizing curves u2 = c by the arc-length s, the parametrization
u1 = s, is a geodetic motion. This it satisfies equations (13). We have
du2
ds = 0, d2u2
ds2 = 0, since u2 = c, and further du1
ds = 1 (the parameter is
arc-length) and d2u1
ds2 = 0. Putting this into (13.1), we obtain
Γ1
11 = 0 , Γ2
11 = 0 .
Since the parametric net is orthogonal, we have g12 = (f1, f2) = 0. Using
the decomposition 12.(1), we obtain
dg11
du1
=
∂(f1, f1)
∂u1
= 2(f1, f11) = 2Γ1
11(f1, f1) = 0 .
83
Differentiating (f1, f2) = 0, we find
0 =
∂(f1, f2)
∂u1
= (f11, f2) + (f1, f12) = Γ2
11(f2, f2) + (f1, f12) = 0 ,
hence (f1, f12) = 0. Now we obtain
∂g11
∂u2
=
∂(f1, f1)
∂u2
= 2(f1, f12) = 0 .
Thus g11 is a constant k > 0.
Consider a curve from p to q given by the parametrization u1 = t,
u2 = f(t), f(0) = 0, f(a) = 0. Its length is equal to
e13.2e13.2 (2)
a
0
k + g22 t, f(t)
df
dt
2
dt .
The length of the geodesic C is
a
0
√
k dt = a
√
k. Since g22 > 0, the integral
(2) is greater or equal to
√
ka and the equality holds obly for df
dt = 0. Using
f(0) = 0, this means that f(t) = 0 for all t.
13.10 13.10 Remark. The oldest approach to the notion of geodesics comes
exactly from the property that these are shortest curves connecting two
points on the surface. Thus differential equations were derived using the
calculus of variation.
13.11 13.11. The curvature κ of a plane curve f(s) satisfies κ = de1
ds , e1 = df
ds .
An analogy of this property is used to define geodetic curvature of the curve
C ⊂ S. Assume that C is parametrized by the arclength γ(s). Then ˙γ = dγ
ds
is a unit vector.
Definition. Geodetic curvature κg of the curve γ(s) on the surface
S is defined via the relation κg = ˙γ
ds .
That is, the geodetic curvature belongs to the inner geometry of the
surface.
Remark. Also a notion of geodetic torsion of a curve on the surface
can be defined but this no more belongs to the inner geometry of the surface.
13.12 13.12. We shall show hot the usual curvature κ and the geodetic curvature
κg of a curve C on the surface S are related.
84
Theorem. Let α be the angle between the normal direction of the surface
and the osculating plane of the curve C ⊂ S. Then κg = κ sin α. Further,
we have κg = 0 in inflection points of the curve C.
Proof. In a non-inflection point, the vector d ˙γ
ds lies in the osculating plane. JS: missing picture
It follows from the picture depictin the section by the plane perpendiculatr
to the osculating plane ω of the curve C that ˙γ
ds = d ˙γ
ds sin α. In
inflection points, we have d ˙γ
ds = o thus also ˙γ
ds = o.
13.13 13.13 Corollary. The curve C on the surface S is geodetic if and only if
κg = 0 at all points of C.
Proof. This follows directly from theorems 13.7 and 13.12.
Lines in a plane are characterized by the property κ = 0. This is one
of analogies between lines on a plane and geodetic curves on a surface.
13.14 13.14. Using geodesics, we can extend certain constructions from the euclidean
planes to surfaces. The simplest example is the notion of geodetic
circles on the surface S.
It follows from properties of solutions of systems of 2nd order differential
equations that for each point p ∈ S, there is a number rp > 0 such that
for each 0 < r < rp and on every geodesic through p there are exactly two
points q1 and q2 (one in each direction) such that the length of the arc
between p and q1 and of the arc between p and q2 is equal to r. Moreover,
it holds that the set K(p, r) of all such points is a curve on S.
Definition. The curve K(p, r) is called geodetic circle of the radius r
with the center p on the surface S.
In the case of the sphere S with the raidus , we shall show that this
constraction works generally onlu locally. If r < π then K(p, r) is a usual
circle on S which is a curve. For r = π , one moves along geodesics in all
directions to the point opposite to p on S. In a sense, putting r = π , the
circle K(p, r) “collapses” into a single point.
13.15 13.15. Geodetic circles have constant curvature in the plane and the same
is true (as we have just shown) on spheres. This is not true in general,
however. Already ellipsoid with axis of different lengths is an example of a
surface where geodetic cicles do not have constant geodetic curvature. (We
shall mention one more statement in this directions later in 14.7.)
85
14 Surfaces with constant Gaussian curvature
The main aim of this chapter is to show a relation between inner geometry
of surfaces with noneuclidean geometries. These observation are not
only geometrically nice but also played an important role in the history of
mathematics. It was mainly understanding of inner geometry which (in the
beginning of the 19. century) motiovated leading geometres to believe that
noneuclidean geometry forms a mathematical theory in the same sense as
euclidean geometry. Proofs in this chapter require either too extensive computations
or tools going far beyond theory built so far. These proofs will
be omitted — we are interested more in a general view than in a detailed
technical matters.
14.1 14.1. We shall investigate local isometries of S with itself. Considering
an arbitrary surface, the locallity assumption is so natural that the word
“local” will be omitted. Consider a surface of revolution S from 8.15 with
the parameter t changed to u. Hence the parametric expression of S is
e14.1e14.1 (1) f(u, v) = g(u) cos v, g(u) sin v, h(u) .
We found g11 = g 2+h 2, g12 = 0, g22 = g2 in 8.15. The surface of revolution
obviously has a 1-parameter system of isometries given by rotations. One
can see that also from the 1. fundamental form
e14.2e14.2 (2) Φ1 = g11(u) du2
+ g22(u) dv2
, g11 = g 2
+ h 2
, g22 = g2
.
The mapping u = ¯u, v = ¯v + c preserves this form since du = d¯u, dv = d¯v.
14.2 14.2. Assume in the opposite direction that the surface S has a 1-parameter
system of isometries such that its trajectories form the system L of curves.
Consider its orthogonal trajectories L . Curves form the systemL transform
to each other by isometries since an isometry preserves angles. Choose
L for coordinate curves u = konst. and L for coordinate curves v = konst.
Our isometries than have the form ¯u = u, ¯v = v + c. The form Φ1 is
preserved by isometries and also g12 = 0 by orthogonality of coordinate
net. Hence
e14.3e14.3 (3) Φ1 = A1(u) du2
+ A2(u) dv2
, A1 > 0 , A2 > 0 .
Change the parameter u to ¯u such that d¯u =
√
A1 du, i.e. ¯u =
√
A1 du
where we keep ¯v = v. Then we have
Φ1 = d¯u2
+ B(¯u) d¯v2
, B > 0 .
86
This is the 1. fundamental form of a surface of revolution given by graph
of the function x = B(z) where we parametrize this curve by arc-length.
Summarizing, we have proved
Theorem. If the surface S has a 1-parameter system of isometries then it
is locally isometric to a surface of revolution.
14.3 14.3. We know that the Gaussian curvature is preserved by isometries.
Hence if the surface S has more then one 1-parameter system of isometries
then the Gaussian curvature must be constant. We know from 11.19 that
a surface is locally isometric to a plane if it has zero Gaussian curvature.
However, this holds more generally: two surfaces with the same constant
Gaussian curvature are locally isometric, see 14.6 below.
The basic example of a surface with zero K is the plane. The basic
example of a surface with a constant positive curvature K = 1
r2 is the
sphere with radius r, see ??. Now we shall present an example of a surface
with a constant negative curvature K = − 1
a2 .
14.4 14.4. We shall need to formula for the Gaussian curvature of the surface
of revolution (1). We computed coefficients of 1. and 2. fundamental form
in 8.15,
e14.4e14.4 (4) K =
h11h22 − h2
12
g11g22 − g2
12
=
h (g h − h g )
g(g 2 + h 2)2
.
Since rotations are isometries, K is independent on the rotation parameter
v.
14.5. A plane curve with the parametric expression
e14.5e14.5 (5) x = a sin u = g(u) , z = a ln tg
u
2
+ cos u = h(u) ,
u ∈ (0, π), u =
π
2
is called traktrix with the parameter a > 0, see the picture. (Given JS: Missing picture
u = π
2 , we have x = a and lim
u→ π
2
h(u) = 0. The point (a, 0) does not lie
on the curve, however. This is a sort of singularity.) Rotation around the
z-axis yields the surface called pseudosphere. Its Gaussian curvature can
be computed using (4). We have g = a cos u, g = −a sin u,
h = a
1
tg u
2 cos2 u
2
·
1
2
−sin u = a
1
sin u
−sin u , h = a −
cos u
sin2
u
−cos u .
87
Putting this to (4), we obtain K = − 1
a2 .
Thus pseudosphere with the parameter a has negative constant Gauassian
curvature − 1
a2 . In a sense, this is “opposite” of the sphere with K = 1
r2
which also motivates the terminology
14.6 14.6. We know that isometries of the plane are Euclidean transformations,
i.e. plane has 3-parameter system of isometries. Isometries of the sphere
S are given by restriction of Euclidean motions in E3 which preserve S.
Geometrically, we can easily see that for each pair of points p, ¯p ∈ S and
each pair of unit perpendicular vectors e1, e2 ∈ TpS and ¯e1, ¯e2 ∈ T¯pS,
there exists a unique isometry g: S → S such that g(p) = ¯p, Tpg(e1) = ¯e1,
Tpg(e2) = ¯e2. A similar statement holds for an arbitrary suraface with a
constant Gaussian curvature. We shall present this without proof.
Theorem (Minding). Let S and ¯S are surfaces with the same constant
Gaussian curvature. Then for each pair of points p ∈ S and ¯p ∈ ¯S and
each pair of unit perpendicular vectors e1, e2 ∈ TpS and ¯e1, ¯e2 ∈ T¯p
¯S,
there exists a unique local isometry g from S to ¯S such that g(p) = ¯p,
Tpg(e1) = ¯e1, Tpg(e2) = ¯e2.
Hence local isometries of every surface with constant Gaussian curvature
form a 3-parameter system of isometries (as in the plane).
14.7 14.7. Consider the geodetic circle K(p, r) on the surface S with a constant
Gaussian curvature. It follows from Theorem 14.6 that there is a
1-parameter system of local isometries on S which preserve the point p (in
a sense, these are rotations around the point p). Considering small r, this
means that the geodetic curvature of the geodetic circle is the same at all
points. The case of the sphere was already mentioned 13.15. One can show
this holds also globally:
Theorem. Geodetic circles on surfaces with constant Gaussian curvature
have constant geodetic curvature.
14.8 14.8. Our next tool will be the Gauss-Bonnet theorem. First we shall
formulate necessary mathematical terminology.
Let C be a curveje with the parametrization f(t), t ∈ I.
Definition. Segment U of the curve C corresponding to a closed interval
[a, b] ⊂ I is the set f(t), t ∈ [a, b].
14.9 14.9. Let D ⊂ R2 be an open set. A simple region Ω ⊂ D is an open,
convex and bounded subset such that also its closure Ω lies in D and its
border ∂Ω is formed by a finite number of segments of curves. JS: missing picture
88
Definition. The subset W ⊂ S is called simple region on the surface
S if there exists such a parametrization f : D → E3 of the surface S such
that W = f(Ω) where Ω is a simple region in D.
Note that one should carefully distinguish between a simple surface and
a simple region on a surface.
14.10 14.10. Let h be a function defined on the curve C. Using the given
parametrization f : I → E3 of the curve C, this is the function h(t): I → R.
Consider the segment U of the curve C which corresponds to the interval
[a, b] ⊂ I. Then we define the integral
U
h ds as
e14.6e14.6 (6)
U
h ds =
b
a
h(t) (f1)2 + (f2)2 + (f3)2 dt .
This definition is independent on the choice of parametrization of the curve.
We define the integral
C
h ds using deompocition of the curve C to
segments.
14.11 14.11. Let h be a function defined on the surface S. Given a parametrization
f : D → E3, this is the function h(u1, u2): D → R. Let Ω ⊂ D be
bounded region such that Ω ⊂ D. PWe write W = f(Ω). Recall that we
introduced the volume element dV = g11g22 − g2
12 du1 du2 of the surface
in ??. The integral
W
h dV is defined by
e14.7e14.7 (7)
W
h dV =
Ω
h(u1, u2) g11g22 − g2
12 du1 du2 .
Also this definition is independent on the choice of a parametrization of the
surface.
14.12 14.12. Now we shall state (without a proof) one of most interesting results
of the inner geometry of surfaces.
Theorem (Gauss-Bonnet theorem). Let W be a simple region on the surface
S whose border is the curve C of the class C2. Let κg be the geodetic
curvature of the curve C and K be the Gaussian curvature of the surface
S. Then
e14.8e14.8 (8)
W
K dV = 2π −
C
κg ds .
89
Example. To get a first experience with this theorem, we shall discuss the
case of the simple region Ω in the plane whose border is the curve C of the
class C2. Its geodetic curvature is the usual curvature and we have K = 0
for the plane. Thus (8) yields
e14.9e14.9 (9)
C
κ ds = 2π .
In the case of the circle f(t) = (r cos t, r sin t), t ∈ [0, 2π), this can be verified
also by a direct computation. We have κ = 1
r , ds = r2 cos2 t + r2 sin2
t dt =
rdt. Thus
C
κ ds =
2π
0
1
r
rdt = 2π .
Observe moreover that we have
t2
t1
κ ds = t2 − t1 in this case.
14.13 14.13. One can extend the Gauss-Bonnet theorem also to some simple
regions whose border is not differentiable because of “vertices”. We shall
discuss the case of the curvilinear triangle on the surface S. In the following
definition, we understand a triangle in R2 as a closed set including its inner
part.
Definition. The set W ⊂ S is called curvilinear triangle on the surface
S if there exists such local parametrization f : D → E3 of the surface
S and such triangle ∆ ⊂ D that W = f(∆).
Thus a curvilinear triangle is a simple region on S. Its edges are segments
U1, U2, U3 of curves on S. Denote by β1, β2, β3 inner angles of its
tangent lines at vertices p1, p2, p3 as on the picture. JS: missing picture
Theorem (Generalised Gauss-Bonnet). It holds
e14.10e14.10 (10)
W
K dV = 2π −
3
i=1
βi −
3
i=1 Ui
κg ds .
Idea of the proof. Let us “smooth” the border of W at the vertex pi
using a geodesic circle with Ci with a small radius r as on the picture. JS: missing picture
Then one can show that lim
r→0
Ci
κg ds = βi. One can formulate this also as
the idea that as a limit, we have the same situation as in the plane which
we discussed at the end of example 14.12. It follows from (8) that be obtain
the formulae (10) as a limit.
90
de14.14 14.14 Definition. A curvilinear triangle W is called geodetic triangle
on the surface S if all its edges are segments of geodetic curves.
This notion is a direct a straightforward modification of the notion of
a triangle for the inner geometry of surfaces.
We κg = 0 in (10) in this case. This shows
Corollary. A geodetic triangle W on the surface S satisfies
e14.11e14.11 (11)
W
K dV = 2π − β1 − β2 − β3 .
14.15 14.15. The following geometrically very interseting statement follows directly
from (11):
Theorem (Special Gauss-Bonnet). If W is a geodetic triangle on a surface
with constant Gauss curvature K of the area V and inner angles α1, α2,
α3 then the following holds:
e14.12e14.12 (12) α1 + α2 + α3 = π + KV .
Proof. PFor K constant, the integral in (11) is equal KV and inner angles
are given by the relation βi = π − αi, i = 1, 2, 3.
14.16. Examples. a) Considering developable surfaces, we have K = 014.16
as in the plane. Then (12) yields the well known theorem about the sum
of angles in a triangle. α1 + α2 + α3 = π.
b) Considering the sphere S with radius r, we have K = 1
r2 . Thus the
sum of angles in a geodetic triangle is greater then π. In fact, it follows
from (12) that the sum af angles minus π is proportional to the area of the
geodetic triangle. It is interesting to realiye that one can compute are of the
sphere from (12). Since geodetic curves on S are great circles, one eighth
of the sphere is a geodetic triangle with all three right angles. Denote by
V its area. Since the Gaussian curvature of the sphere with radius r is 1
r2 ,
it follows from (12) that 3π
2 = π + 1
r2 V , i.e. V = 1
2πr2.
14.17 14.17. The 5th Euclid axiom states that for a given line p in the plane,
there is a unique line nonintersecting p through every point A /∈ p. Nowadays,
this is known as the Euclid axiom about parallel lines. This statement
is so essentially different from other Euclid axioms that many mathematicians
for centuries tried to rpove that the 5th axiom is a consequence of
remaining Euclid axioms. However, it was not possible to prove, despite of
many (incorrect) attempts, to show that the opposite of the Euclid axiom
about parallel lines leads to a contradiction.
91
Thus one of consequences of negation of the 5. axiom, i.e. an assumption
of existence at least two lines through the point A nonitersecting p,
is the statement that the sum Σof angles in a triangle is smalles than π.
Moreover, the difference π−Σ, so called angle defect, is propotional to the
area of triangle. Such statement seemed absurd to many mathematicians.
The special Gauss-Bonnet theorem shows this case happens in the inner
geometry of a surface with negatiove constant Gaussian curvature. Also
our theorem 14.6 about the three-parameter system of local isometries on
a surface with constant Gaussian curvature corresponds to Euclidean transformations
in the classical geometry of plane – with the Euklid axiom about
parallel lines or with its negation. Precise constructions of noneuclidean
geometries were found in the 2nd half of the 19. century (using essentially
projective geometry tools).
14.18. Negation of the 5. axiom leads only to one type of noneuclidean geometries
which are called Lobachevsky geometries or hyperbolic geometries.
These correspond to surfaces with negative constant Gaussian
curvature. However, D. Hilbert shown in the year 1901 that Lobachevsky
plane cannot be globaly realized on a surface in E3. (This is the essential
explanation why the tractrix in ??, whose rotation yiedls the pseudosphere,
has a singular point.
An important analogy between properties of surfaces with a posotive
and negative constant Gaussian curvature, stated in particular in theorems
14.3, 14.6 and 14.15, motivated to include also elliptic geometries (often
related to the name B. Riemann) into noneuclidean geometries. These correspond
to the inner geometry of surfaces with positive constant Gaussian
curvature. Let us not the notion of Riemannian geometry is usually
using in a meaning different from elliptic geometries. In fact, Riemannain
geometry denotes far more important theory, which B. Riemann iniciated,
and which is nowadays one of more important parts of differential geometry,
see e.g. the textbook [5] for a basic information.
92
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1 [1] J. Bureˇs, K. Hrubk, Diferentiaci´aln geometrie kˇrivek a ploch, Skriptum,
UK Praha, 1998.
2 [2] M. P. Do Carmo, Differential geometry of Curves and Surfaces,
Prentice-Hall, New Jersey, 1976.
3 [3] M. Doupovec, Diferenci´aln geometrie a tenzorov´y poˇcet, Skriptum,
VUT Brno, 1999.
4 [4] W. Klingenberg, A Course in Differential Geometry, Springer-Verlag,
1978.
5 [5] I. Kol´aˇr, ´Uvod do glob´aln anal´yzy, Skritpum, MU Brno, 2003.
6 [6] A. Vanˇzurov´a, Diferenci´aln geometrie kˇrivek a ploch, Skriptum, PU
Olomouc, 1996.
93