{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Distance between point and line" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Eukleidian metric. . . d istance between two points:" }{MPLTEXT 1 0 47 "\nrho:=(X,Y)->sqrt((X[1 ]-Y[1])^2+(X[2]-Y[2])^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoGj+ 6$%\"XG%\"YG6\"6$%)operatorG%&arrowGF)-%%sqrtG6#,&*$),&&9$6#\"\"\"F7&9 %F6!\"\"\"\"#F7F7*$),&&F56#F;F7&F9F@F:F;F7F7F)F)F)6$\"\"!FC" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Line, which is not vertical has th e form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "L:=x->p*x+q;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGj+6#%\"xG6\"6$%)operatorG%&arrow GF(,&*&%\"pG\"\"\"9$F/F/%\"qGF/F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 58 "So the points, which lies on the line has the coordinat es:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "l:=[x,L(x)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG7$%\"xG,&*&%\"pG\"\"\"F&F*F*%\"q GF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The distance between gener al point X and point l on the line L with first coordinte x is:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sigma:=rho(X,l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG*$,4*$)&%\"XG6#\"\"\"\"\"#F,F,*(F-F ,F)F,%\"xGF,!\"\"*$)F/F-F,F,*$)&F*6#F-F-F,F,**F-F,F5F,%\"pGF,F/F,F0*(F -F,F5F,%\"qGF,F0*&)F8F-F,F2F,F,**F-F,F8F,F/F,F:F,F,*$)F:F-F,F,#F,F-" } }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "sigma;" "6#%&sigmaG" }{TEXT -1 20 " is the function of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 61 ". We are looking for x which realize minimum of the distance:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Dsigma:=simplify(diff(sigma, x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'DsigmaG*&,,&%\"XG6#\"\"\"! \"\"%\"xGF**&&F(6#\"\"#F*%\"pGF*F+*&)F1F0F*F,F*F**&F1F*%\"qGF*F*F*,4*$ )F'F0F*F**(F0F*F'F*F,F*F+*$)F,F0F*F**$)F.F0F*F***F0F*F.F*F1F*F,F*F+*(F 0F*F.F*F5F*F+*&F3F*F;F*F***F0F*F1F*F,F*F5F*F**$)F5F0F*F*#F+F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x_min:=solve(Dsigma=0,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&x_minG,$*&,(&%\"XG6#\"\"\"!\"\"*&& F)6#\"\"#F+%\"pGF+F,*&F1F+%\"qGF+F+F+,&F+F+*$)F1F0F+F+F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "so the distance btween the point and the line which is minimum of distances between the point and all of the \+ point on the line is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xx x:=simplify(subs(x=x_min,sigma));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$xxxG*$*&,(*&&%\"XG6#\"\"\"F,%\"pGF,F,%\"qGF,&F*6#\"\"#!\"\"F1,&F,F,* $)F-F1F,F,F2#F,F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P:=una pply(xxx,X,p,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGj+6%%\"XG%\" pG%\"qG6\"6$%)operatorG%&arrowGF**$*&,(*&&9$6#\"\"\"F59%F5F59&F5&F36# \"\"#!\"\"F:,&F5F5*$)F6F:F5F5F;#F5F:F*F*F*6%\"\"!FAFA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P([1,2],1,1);\nP([1,1],1,1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"F%#\"\"\"F%F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Now we have:" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 48 "the points X (t he number of points is N=nops(X))" }}{PARA 15 "" 0 "" {TEXT -1 27 "the values Y in this points" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "and e are suming the squer of distnces between the points " }{XPPEDIT 18 0 "[X[i], Y[i]];" "6#7$&%\"XG6#%\"iG&%\"YG6#F'" }{TEXT -1 30 " and the l ine with parameters " }{XPPEDIT 18 0 "p,q;" "6$%\"pG%\"qG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "R:=simplify(\nsum(P ([X[i],Y[i]],p,q)^2,i=1..N)\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"RG*&,&*&%\"NG\"\"\")%\"qG\"\"#F)F)-%$sumG6$,,*&)&%\"XG6#%\"iGF,F))% \"pGF,F)F)**F,F)F3F)F8F)F+F)F)**F,F)F3F)F8F)&%\"YGF5F)!\"\"*(F,F)F+F)F ;F)F=*$)F;F,F)F)/F6;F)F(F)F),&F)F)*$F7F)F)F=" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 59 "and we are looking for p and q, which realize minimum o f R:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "xxx:=solve(\{\ndiff (R,p)=0,\ndiff(R,q)=0\n\},\n\{p,q\});\nzzz:=allvalues(xxx);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$xxxG6$<$/%\"pG$!+=AyK6!\"*/%\"qG$\"+Yb&>L &F+<$/F-$\"+OXWIH!#5/F($\"+&=Ay#))F4" }}{PARA 8 "" 1 "" {TEXT -1 72 "E rror, (in allvalues) invalid option \{q = .2930444536, p = .8827822185 \}\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "X:=[1,2,3,4];\nY:=[ 1.1,2.2,2.9,3.8];\nN:=nops(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" XG7&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG7&$ \"#6!\"\"$\"#AF($\"#HF($\"#QF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" NG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "with(plots):\nA :=pointplot([seq([X[i],Y[i]],i=1..N)]):\nline[1]:=subs(xxx[1],p*x+q); \nline[2]:=subs(xxx[2],p*x+q);\nB:=plot(\{line[1],line[2]\},x=0..5):\n display(\{A,B\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%lineG6#\"\"\" ,&*&$\"+=AyK6!\"*F'%\"xGF'!\"\"$\"+Yb&>L&F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%lineG6#\"\"#,&*&$\"+&=Ay#))!#5\"\"\"%\"xGF-F-$\"+OX WIHF,F-" }}{PARA 13 "" 1 "" {GLPLOT2D 276 276 276 {PLOTDATA 2 "6'-%'CU RVESG6$7S7$$\"\"!F)$\"3l*****fab>L&!#<7$$\"3GLLL3x&)*3\"!#=$\"3bw\\y-% )\\3_F,7$$\"3umm\"H2P\"Q?F0$\"3(HM&f-!z55&F,7$$\"3MLL$eRwX5$F0$\"3/ZxV _YF!)\\F,7$$\"33ML$3x%3yTF0$\"3%)Q.jK&p'e[F,7$$\"3emm\"z%4\\Y_F0$\"30. -t!QUwt%F,7$$\"3`LLeR-/PiF0$\"3d`\\O=ZVDYF,7$$\"3]***\\il'pisF0$\"32%> 2L=]#4XF,7$$\"3>MLe*)>VB$)F0$\"3]jn%4(>4*Q%F,7$$\"3Y++DJbw!Q*F0$\"3Sns \\2\">$pUF,7$$\"3%ommTIOo/\"F,$\"3=gJ[&*z6YTF,7$$\"3YLL3_>jU6F,$\"3Dr8 '*)Q-w.%F,7$$\"37++]i^Z]7F,$\"33R$py^Ra\"RF,7$$\"33++](=h(e8F,$\"3qN9H M]x#z$F,7$$\"3/++]P[6j9F,$\"35S!)=v]cuOF,7$$\"3UL$e*[z(yb\"F,$\"3i(o0w 5>sc$F,7$$\"3wmm;a/cq;F,$\"3[&*fUpVdRMF,7$$\"3%ommmJF,$\"3-2$f[??V 4$F,7$$\"3K+]i!f#=$3#F,$\"3aDG\\]L;sHF,7$$\"3?+](=xpe=#F,$\"35$Gcv8Te& GF,7$$\"37nm\"H28IH#F,$\"3mh%e86rWt#F,7$$\"3um;zpSS\"R#F,$\"3RoroRb,BE F,7$$\"3GLL3_?`(\\#F,$\"3i\\\\^[cz-DF,7$$\"3fL$e*)>pxg#F,$\"3=&*=xo4#z P#F,7$$\"33+]Pf4t.FF,$\"3M>R'H><#pAF,7$$\"3uLLe*Gst!GF,$\"3W_S0PT\"=:# F,7$$\"30+++DRW9HF,$\"3ut=S#HD0.#F,7$$\"3:++DJE>>IF,$\"3NoLg?y'=\">F,7 $$\"3F+]i!RU07$F,$\"3W*y%Q:11(z\"F,7$$\"3+++v=S2LKF,$\"31,A3)y'ep;F,7$ $\"3Jmmm\"p)=MLF,$\"3G:Li%)e/b:F,7$$\"3B++](=]@W$F,$\"3\\LY\">-\\FV\"F ,7$$\"35L$e*[$z*RNF,$\"3<$y7u))H>K\"F,7$$\"3e++]iC$pk$F,$\"3uYIB-`x+7F ,7$$\"3[m;H2qcZPF,$\"3]BlPz#yn3\"F,7$$\"3O+]7.\"fF&QF,$\"3'*4PUPa=w'*F 07$$\"3Ymm;/OgbRF,$\"3D))GV<7=6&)F07$$\"3w**\\ilAFjSF,$\"3]pU2m(H:H(F0 7$$\"3yLLL$)*pp;%F,$\"3^CCSf/'o6'F07$$\"3)RL$3xe,tUF,$\"3M[K3K9f:\\F07 $$\"3Cn;HdO=yVF,$\"3nGz\")=&pUs$F07$$\"3a+++D>#[Z%F,$\"3p%pMkSo&HEF07$ $\"3SnmT&G!e&e%F,$\"3)oL'o%R<\\P\"F07$$\"3#RLLL)Qk%o%F,$\"3y=KvHdUFD!# >7$$\"37+]iSjE!z%F,$!3?eJR9%)HP%*Fiy7$$\"3a+]P40O\"*[F,$!3[KUZ3m!*)3#F 07$$\"\"&F)$!3!Q+++Wb&>LF0-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7S7$F($ \"3x*****f`W/$HF07$F.$\"3gD=hT:b#*QF07$F4$\"35rk$Gqv'HZF07$F9$\"3?%[;U P46n&F07$F>$\"3z9t2zMy=mF07$FC$\"3-V?MaM&>c(F07$FH$\"3hYJdbFRO%)F07$FM $\"3\"o!p\\-S#=M*F07$FR$\"3[C)H$HA#y-\"F,7$FW$\"3c?0Ha<;@6F,7$Ffn$\"30 \"f)fGH<<7F,7$F[o$\"3'46*=B'RF,7$Fcq$\"3S)p6GiVo.#F,7$Fhq $\"3;:D\\-5/K@F,7$F]r$\"3b,z;?9pAAF,7$Fbr$\"3kx)e6ivsJ#F,7$Fgr$\"3a;1m VM8/CF,7$F\\s$\"3m#)HJR8#y\\#F,7$Fas$\"3]0CRKn8&f#F,7$Ffs$\"37'>W\"o+& )zEF,7$F[t$\"3g*z`:zU8x#F,7$F`t$\"3cZ`SFP'e'GF,7$Fet$\"3eFPpASLeHF,7$F jt$\"3EJ#>\")y.y/$F,7$F_u$\"3!e$oW3Z9ZJF,7$Fdu$\"387rsVpSOKF,7$Fiu$\"3 .T9`KMrJLF,7$F^v$\"3,Bwrw_2=MF,7$Fcv$\"3Y=ac$e\"\\7NF,7$Fhv$\"3(GMu-(* H8g$F,7$F]w$\"3EeF+#o\">%p$F,7$Fbw$\"3!f%Gzy4)\\y$F,7$Fgw$\"3gezhe*G+) QF,7$F\\x$\"3M3*4*f9drRF,7$Fax$\"3G\\tD*)o=lSF,7$Ffx$\"3k\"Q%eNr-eTF,7 $F[y$\"3^TR%*zwLVUF,7$F`y$\"3;*)*p5>86M%F,7$Fey$\"3Tc9\"Qxk&GWF,7$F[z$ \"3s9G)4S1=_%F,7$F`z$\"3&QN\\b`]5h%F,7$Fez$\"3b++5Yb&pq%F,-Fjz6&F\\[lF (F][lF(-%'POINTSG6&7$$\"\"\"F)$\"#6F_[l7$$\"\"#F)$\"#AF_[l7$$\"\"$F)$ \"#HF_[l7$$\"\"%F)$\"#QF_[l-%+AXESLABELSG6$Q\"x6\"Q!Fcfl-%%VIEWG6$;F(F ez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(xxx[1],R);\nsubs(xxx[2],R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yCed\"!# 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "23 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }