GRAFIKA V MAPLU -- I. CAST
Uvod
Velikonocni vajicko
> with(plots):setoptions3d(scaling=constrained,project
ion=.5,style=patchnogrid):
Warning, the name changecoords has been redefined
> X:=cos(x)*cos(y)*(1+.2*sin(y)):
Y:=sin(x)*cos(y)*(1+.2*sin(y)):
Z:=1.5*sin(y):
a:=plot3d([X,Y,Z],x=-Pi..Pi,y=-Pi/2..Pi/2,color=[1,.
8,.5],ambientlight=[.4,.4,.4],light=[75,50,.8,.8,.7]
,grid=[30,30]):
p:=t+.1*sin(10*t):
q:=.5*cos(10*t):
X0t:=cos(p)*cos(q-.3)*(1+.2*sin(q-.3)):
Y0t:=sin(p)*cos(q-.3)*(1+.2*sin(q-.3)):
Z0t:=1.5*sin(q-.3):
X1t:=cos(p)*cos(q-.6)*(1+.2*sin(q-.6)):
Y1t:=sin(p)*cos(q-.6)*(1+.2*sin(q-.6)):
Z1t:=1.5*sin(q-.6):
X2t:=cos(p)*cos(q+.1)*(1+.2*sin(q+.1)):
Y2t:=sin(p)*cos(q+.1)*(1+.2*sin(q+.1)):
Z2t:=1.5*sin(q+.1):
X3t:=cos(p)*cos(q+.6)*(1+.2*sin(q+.6)):
Y3t:=sin(p)*cos(q+.6)*(1+.2*sin(q+.6)):
Z3t:=1.5*sin(q+.6):
X4t:=cos(p)*cos(q-1)*(1+.2*sin(q-1)):
Y4t:=sin(p)*cos(q-1)*(1+.2*sin(q-1)):
Z4t:=1.5*sin(q-1):
b:=tubeplot({[X0t,Y0t,Z0t,radius=.04,color=[.9,.8,.2
]],[X1t,Y1t,Z1t,radius=.03,color=[.0,.7,1]],[X2t,Y2t
,Z2t,radius=.05,color=[.3,.9,.2]],[X3t,Y3t,Z3t,radiu
s=.05,color=[.1,.6,.6]],[X4t,Y4t,Z4t,radius=.025,col
or=[.8,.3,.2]]},t=-Pi..Pi,numpoints=300,tubepoints=7
):
display3d({a,b},orientation=[0,-120]);
> restart;
> f:=x->exp(-x^2)*sin(Pi*x^3);
:=f x e
( )-x
2
( )sin  x3
> plot(f, -2..2);
­0.4
­0.2
0
0.2
0.4
­2 ­1 1 2
Prikaz pro nakresleni funkce plot(f, a..b, options), a..b interval na ose x, options
jsou nepovinne.
Pri kresleni formule je treba pouzit plot(f(x), x=a..b, options).
> plot(f(x), x=-2..2);
­0.4
­0.2
0
0.2
0.4
­2 ­1 1 2
x
> plot(f, -infinity..infinity);
-infinity infinity
V tomto pripade Maple transformuje realnou osu na interval (-1,1).
Vice funkci v jednom obrazku:
> plot({f(x), exp(-x^2), -exp(-x^2)}, x=-2..2);
­1
­0.5
0.5
1
­2 ­1 1 2
x
> plot([sin(x), cos(x)], x=0..2*Pi, color=[red, blue],
thickness=[2,3], legend=["sin", "cos"]);
sin
cos
­1
­0.5
0
0.5
1
1 2 3 4 5 6
x
Pokud chceme omezit rozsah zobrazovanych hodnot na ose y, musime to
Maplu sdelit:
> plot(sin(x^2)/x^2, x=-6..6, y=0..1);
0
0.2
0.4
0.6
0.8
1
y
­6 ­4 ­2 2 4 6
x
Maple voli meritko na obou osach tak, aby obrazek co nejlepe "zaplnil"
display. Pokud chceme stejne meritko na osach, pouzijeme volbu
scaling=constrained.
> plot(sin(x^2)/x^2, x=-6..6, scaling=constrained);
0
1
­6 ­4 ­2 2 4 6
x
Nekdy je vymezeni rozsahu zobrazovanych hodnot nutne:
> plot(tan, -Pi..Pi);
­1500
­1000
­500
0
­3 ­2 ­1 1 2 3
> with(plots):
Warning, the name changecoords has been redefined
> display(%%, view=[-4..4,-10..10]);
­10
­8
­6
­4
­2
0
2
4
6
8
10
­4 ­3 ­2 ­1 1 2 3 4
> plot(tan, -Pi..Pi, -10..10);
­10
­8
­6
­4
­2
0
2
4
6
8
10
­3 ­2 ­1 1 2 3
Zamezeni "spojovani" nespojitych funkci provedeme pomoci volby
discont=true.
> plot(tan(x), x=-Pi..Pi, y=-10..10, discont=true);
­10
­8
­6
­4
­2
0
2
4
6
8
10
y
­3 ­2 ­1 1 2 3
x
> plot(tan(x), x=-Pi..Pi, -10..10, discont=true,
xtickmarks=[-3.14=`-Pi`, -1.57=`-Pi/2`, 1.57=`Pi/2`,
3.14=`Pi`]);
­10
­8
­6
­4
­2
0
2
4
6
8
10
-Pi -Pi/2 Pi/2 Pi
x
> plot(sin(x), x=0..20, `y(x)`=-0.5..1, xtickmarks=8,
ytickmarks=4, title=`Graf funkce sinus`);
Graf funkce sinus
­0.4
­0.2
0
0.2
0.4
0.6
0.8
1
y(x)
2 4 6 8 10 12 14 16 18 20
x
> plot(sin(x), x=0..20, `y(x)`=-0.5..1,
xtickmarks=[5,10,15,20], ytickmarks=[-0.4,0,1],
title=`Graf funkce sinus`);
Graf funkce sinus
­0.4
0
1
y(x)
5 10 15 20
x
> with(plots):
> plot1:=plot(sin, 0..20, -0.5..1, tickmarks=[5,9],
title= `Graf funkce sinus`):
> plot2:=textplot({[4,0.75,`sin(x)`]}, align={ABOVE,
RIGHT}):
Implicitni nastaveni je centrovani horozontalni i vertikalni. Zarovnani menime
parametrem align=t, kde t muze byt BELOW,RIGHT,ABOVE a LEFT.
> display({plot1, plot2});
Graf funkce sinus
sin(x)
­0.5
­0.4
­0.3
­0.2
­0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16 18 20
Pro znazornovani dat se pouziva style=point.
> plot(sin, 0..2*Pi, scaling=constrained, style=point);
­1
­0.5
0
0.5
1
1 2 3 4 5 6
> plot(sin, 0..2*Pi, scaling=constrained, style=point,
symbol=circle);
­1
­0.5
0
0.5
1
1 2 3 4 5 6
> plotpoints:=[seq([i, ithprime(i)], i=1..9)];
plotpoints [ ],1 2 [ ],2 3 [ ],3 5 [ ],4 7, , , ,[:=
[ ],5 11 [ ],6 13 [ ],7 17 [ ],8 19 [ ],9 23, , , , ]
> plot(plotpoints, style=point, symbol=box);
5
10
15
20
2 4 6 8
> pointplot(plotpoints);
5
10
15
20
2 4 6 8
>
> plots[polygonplot]([[3,-2],[7,-2],[5,5]],color=grey,ax
es=framed);
­2
­1
0
1
2
3
4
5
3 4 5 6 7
> ?plot[options]
Styl a tloustka car
> for i from 0 to 15 do
thick[i]:=plot(i, x=0..1, thickness=i):
od:
> display(convert(thick, set), axes=box,
tickmarks=[0,15], title=`thickness option:
[0,1,...,15]` );
thickness option: [0,1,...,15]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
> for i from 1 to 4 do
line[i]:=plot(i, x=0..1, linestyle=i):
od:
> display(convert(line, set), axes=box, tickmarks=[1,4],
title=`linestyle option: [1,2,3,4]`);
linestyle option: [1,2,3,4]
1
2
3
4
0.5
x
> ?linestyle
> interactive(sin(x), x);
Initializing Java runtime environment.
Struktura dvoj-dimenzionalni grafiky
Vytvareni obrazku probiha ve dvou fazich:
1) Jsou spocitany funkcni hodnoty v referencnich bodech a tyto jsou ulozeny do
objektu datoveho typu PLOT.
2) Objekt je vykreslen na obrazovce.
> PLOT(CURVES([[1,1],[2,2], [3,1], [1,1]]),
AXESSTYLE(NONE), SCALING(CONSTRAINED));
> P:=plot([[1,1], [2,2], [3,1], [1,1]], axes=none,
scaling=constrained);
P PLOT CURVES((:=
[ ], , ,[ ],1. 1. [ ],2. 2. [ ],3. 1. [ ],1. 1. ,
( )COLOUR , , ,RGB 1.0 0. 0. ),
( )SCALING CONSTRAINED ,
( )AXESSTYLE NONE ,
( )AXESLABELS ,"" "" ,
( )VIEW ,DEFAULT DEFAULT )
> P;
> f:=x->sqrt(2)-sqrt(2*sqrt(x)):
> P:=plot(f(x), x=0..1, y=0..sqrt(2));
P PLOT CURVES [((:=
[ ],0. 1.41421356237309515 [,
0.000681161067708333304,
1.18574447219701806] [,
0.00136232213541666661,
1.14251649477750617] [,
0.00204348320312499991,
1.11353133002946070] [,
0.00272464427083333321,
1.09110947646304536] [,
0.00408696640624999982,
1.05664011231514365] [,
0.00544928854166666643,
1.02997588452241362] [,
0.00817393281249999965,
0.988984671428109086] [,
0.0108985770833333329,
0.957275382020941201] [,
0.0163478656249999993,
0.908528339756601633] [,
0.0217971541666666657,
0.870819427181917072] [,
0.0312799478125000002,
0.819467599431483952] [,
0.0407627414583333348,
0.778764352651153735] [,
0.0620915279166666667,
0.708264957079369761] [,
0.0835616954166666648,
0.653857857823032074] [,
0.104929818958333323,
0.609317637273809498] [,
0.124740804791666660,
0.573753399132445496] [,
0.145253933124999984,
0.541147768783589056] [,
0.166468639791666651,
0.510880084658935751] [,
0.187615310624999976,
0.483465658085454830] [,
0.209367260833333346,
0.457587539678707467] [,
0.228526390416666670,
0.436415690230224751] [,
0.250095032499999981,
0.414118543416856166] [,
0.271752237499999993,
0.393137064899568456] [,
0.292622967499999997,
0.374072838367068172] [,
0.311575589791666663,
0.357625123681850176] [,
0.334112090833333320,
0.339016557617175084] [,
0.353203463333333301,
0.323975760718647754] [,
0.375411268124999975,
0.307228338718146032] [,
0.395068938333333286,
0.293013226572828689] [,
0.416636518125000022,
0.278014749677451211] [,
0.437173954375000006,
0.264264594371818040] [,
0.458602614583333345,
0.250424880036826503] [,
0.478280813958333329,
0.238136613844476264] [,
0.499506410416666658,
0.225300046202927761] [,
0.521553839791666585,
0.212392572803464352] [,
0.540746191874999949,
0.201485673008004351] [,
0.561474457916666614,
0.190027307452930394] [,
0.582888784999999965,
0.178518183114692697] [,
0.603838526249999896,
0.167561664675841993] [,
0.624108478124999921,
0.157228784514846343] [,
0.646614803749999911,
0.146046676928994090] [,
0.666837738333333263,
0.136245382971698303] [,
0.688430037499999980,
0.126023481864875775] [,
0.707995869791666620,
0.116966550869118891] [,
0.729386492499999984,
0.107277243374912290] [,
0.749513401458333228,
0.0983530679416697584] [,
0.770551820624999939,
0.0892148367826661737] [,
0.791120720833333291,
0.0804597072567572180] [,
0.812654453124999954,
0.0714749504508052080] [,
0.833393996666666692,
0.0629888177456983112] [,
0.854603175416666638,
0.0544727801778668308] [,
0.875636731458333317,
0.0461823989272109880] [,
0.894964384999999973,
0.0386950615003713860] [,
0.917116057083333325,
0.0302614205047448959] [,
0.936928776666666629,
0.0228467429151371082] [,
0.958053268125000023,
0.0150695701281506889] [,
0.978272101875000023,
0.00774536984891516233] [ ],1. 0., ],
( )COLOUR , , ,RGB 1.0 0. 0. ),
( )AXESLABELS ,"x" "y" ,
( )VIEW ,..0. 1. ..0. 1.414213562 )
Nejdrive se vyhodnoti prvni argument prikazu plot (popisuje zadanou funkci),
dale jsou vybrany equidistantni body ze zadaneho intervalu (implicitne 49
bodu) a numericky jsou v techto bodech spocteny funkcni hodnoty. Dale se
maple diva na tyto body, jako by byly spojeny useckami a kontroluje, jestli
mezi nekterymi useckami neni prilis velky uhel. Pokud ano, prida do teto
oblasti dalsi referencni body. Maximalni pocet referencnich bodu je urcovan
promennou resolution, jejiz implicitni hodnota je 200. Tedy pocet referencnich
bodu je cele cislo z intervalu [49,200].
Pomoci funkce replot z balicku plots muzeme zvetsovat ci zmensovat drive
nakresleny obrazek a ilustrovat vyse uvedenou skutecnost.
> plots[replot] (P, x=0..0.01, y=1..sqrt(2));
1
1.1
1.2
1.3
1.4
y
0 0.002 0.004 0.006 0.008 0.01
x
> infolevel[plot] := 2:
> plot(f(x), x=0..1, y=0..sqrt(2)):
plot/adaptive: evalhf succeeded
plot/adaptive: produced 59. output segments
plot/adaptive: using 59. function evaluations
> plot(f(x), x=0..1, y=0..sqrt(2), adaptive=false):
plot/adaptive: evalhf succeeded
plot/adaptive: produced 49. output segments
plot/adaptive: using 49. function evaluations
> infolevel[plot]:=1:
> plot(x^2, x=0..1, sample=[0,1/2,1], adaptive=true);
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
x
> plot(x^2, x=0..1, sample=[0,1/2,1], adaptive=false);
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
x
> plot(x/(1-cos(5*x)), x=-5..5, -5..5);
­4
­2
0
2
4
­4 ­2 2 4
x
> plot(x/(1-cos(5*x)), x=-5..5, -5..5, numpoints=200);
­4
­2
0
2
4
­4 ­2 2 4
x
> plot(x+sin(2*Pi*x), x=0..49);
0
10
20
30
40
50
10 20 30 40
x
> plot(x+sin(2*Pi*x), x=0..49, numpoints=200);
0
10
20
30
40
50
10 20 30 40
x
Pokud chceme nejakou option zmenit pro celou session, musime jeji hodnotu
nastavit pomoci procedury setoptions.
> setoptions(scaling=constrained);
> setoptions(numpoints);
49
> ?plot[structure]
Specialni dvoj-dimenzionalni obrazky
Funkce dana parametricky:
syntaxe: plot([f(t), g(t), t=a..b], options)
> plot([t*cos(2*Pi*t), t*sin(2*Pi*t), t=0..10],
numpoints=500, scaling=constrained, axes=none);
> plot([cos, sin, 0..2*Pi], scaling=constrained);
­1
­0.5
0.5
1
­1 ­0.5 0.5 1
Krivka v polarnich souradnicich:
polarplot(r-expr,angle=range)
> plots[polarplot](sin(3*theta), theta=0..2*Pi);
­1
­0.8
­0.6
­0.4
­0.2
0
0.2
0.4
­0.8 ­0.6 ­0.4 ­0.2 0.2 0.4 0.6 0.8
polarplot([r-expr, angle-expr, parameter=range)
> plots[polarplot]([sin(t), cos(t), t=0..2*Pi]);
­0.4
­0.2
0.2
0.4
­1 ­0.5 0.5 1
> plot([sin(t)/t, t, t=-6*Pi..6*Pi], coords=polar,
numpoints=250);
­0.6
­0.4
­0.2
0
0.2
0.4
0.6
­0.2 0.2 0.4 0.6 0.8 1
Krivka dana implicitne rovnici
> implicitplot(x^3+y^3-5*x*y+1/5=0, x=-3..3, y=-3..3,
grid=[50,50]);
­3
­2
­1
1
2
y
­3 ­2 ­1 1 2
x
Maple pracuje se zadanou rovnici jako s funkci dvou promennych a generuje
vrstevnici teto funkce na hladine z=0 (rez rovinou z=0).
Nevyhodou teto metody je, ze ziskane obrazky mohou byt "kostrbate" v okoli
singularit a mistech, kde krivka protina sebe samu.
> implicitplot(2*x^4+y^4-3*x^2*y-2*y^3+y^2, x=-5/2..5/2,
y=-5/2..5/2,scaling=constrained);
0.5
1
1.5
2
y
­1.5 ­1 ­0.5 0 0.5 1 1.5
x
> v:=algcurves[parametrization](2*x^4+y^4-3*x^2*y-2*y^3+
y^2,x,y,t);
v


:=
- - - - +8470 6883 t 1548 t2
15 t3
18 t4
+ + + +18 t4
336 t3
5056 t2
26696 t 43257
,
+ + + +4900 6020 t 2689 t2
516 t3
36 t4
+ + + +18 t4
336 t3
5056 t2
26696 t 43257



> plot([op(v), t=-infinity..infinity],
scaling=constrained);
0
0.5
1
1.5
2
­1.5 ­1 ­0.5 0.5 1 1.5
> restart;
Prokladani krivek (linearni regrese, metoda nejmensich ctvercu) - pouzijeme
balik statistic:
> datax:=seq(i, i=1..9);
:=datax , , , , , , , ,1 2 3 4 5 6 7 8 9
> datay:=seq(ithprime(i), i=1..9);
:=datay , , , , , , , ,2 3 5 7 11 13 17 19 23
> with(stats); Digits:=5;
anova describe fit importdata random, , , , ,[
statevalf statplots transform, , ]
:=Digits 5
> aproximace:=fit[leastsquare[[x,y], y=a*x+b,
{a,b}]]([[datax],[datay]]);
:=aproximace =y -
27 x
10
43
18
> pair:=(x,y)->[x,y];
plotpoints:=zip(pair, [datax],[datay]);
:=pair ( ),x y [ ],x y
plotpoints [ ],1 2 [ ],2 3 [ ],3 5 [ ],4 7, , , ,[:=
[ ],5 11 [ ],6 13 [ ],7 17 [ ],8 19 [ ],9 23, , , , ]
> plot1:=plot(plotpoints, style=point):
> plot2:=plot(rhs(aproximace), x=0..10):
> plots[display]({plot1,plot2});
0
5
10
15
20
25
2 4 6 8 10
x
Nasledujici obrazek ukazuje oblast, vyhovujici nerovnicim x+y<5, 0<x, x<=4.
> plots[inequal]({x+y<5, 0<x, x<=4}, x=-1..5, y=-10..10,
optionsexcluded=(color=yellow));
­10
­5
5
10
­1 1 2 3 4 5
Ukladani grafiky
Ulozeni obrazku do postscriptu:
> plot(sin(x), x=-Pi..Pi);
­1
­0.5
0.5
1
­3 ­2 ­1 1 2 3
x
> G:=%:
Nejprve prikazem
> plotsetup(cps, plotoutput="sin.eps", plotoptions=
"portrait,noborder,leftmargin=0,bottommargin=0");
Maplu sdelime, ze ma graficky vystup ukladat do postscriptoveho souboru
sin.eps. Pote vygenerujeme obrazek, jehoz vystup se neobjevi na obrazovce, ale
ulozi do souboru sin.eps v aktulanim adresari.
> G;
Zamezit kresleni ramecku kolem obrazku muzeme pouzitim
doplnujiciho parametru plotoptions=noborder. Vystup
grafiky zpet do zapisniku vratime prikazem:
> plotsetup(default);
> ?plot[device]
>
>