{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 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 262 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 284 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{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 "Hea ding 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 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 Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 62 "The Bernstein basis polynomials \+ and de Casteljau's algorithm " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pete r Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version : 26.3.2007\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "load interpolation and function approximation procedur es" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 284 10 "f cnapprx.m" }{TEXT -1 37 " contains the code for the procedure " } {TEXT 0 9 "casteljau" }{TEXT -1 25 " used in this worksheet. " }} {PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives it s location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\ \\Maple/procdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Bernstein basis polynomials " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 52 "T he Bernstein polynomial associated with a function " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = Sum(f(k/n)*matrix([[n], [k]])*(1-x)^(n-k) *x^k,k = 0 .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$SumG6$**-%\"fG6#*&%\"kG\" \"\"F'!\"\"F2-%'matrixG6#7$7#F'7#F1F2),&F2F2F(F3,&F'F2F1F3F2)F(F1F2/F1 ;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "has the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(f(k/n)*B[k,n ](x),k = 0 .. n);" "6#-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"%\"nG!\"\"F,-&% \"BG6$F+F-6#%\"xGF,/F+;\"\"!F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "B[ k,n](x) = matrix([[n], [k]])*(1-x)^(n-k)*x^k;" "6#/-&%\"BG6$%\"kG%\"nG 6#%\"xG*(-%'matrixG6#7$7#F)7#F(\"\"\"),&F3F3F+!\"\",&F)F3F(F6F3)F+F(F3 " }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "matrix([[n], [k]])=n!/(k!*(n-k) !)" "6#/-%'matrixG6#7$7#%\"nG7#%\"kG*&-%*factorialG6#F)\"\"\"*&-F.6#F+ F0-F.6#,&F)F0F+!\"\"F0F7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The polynomial " }{XPPEDIT 18 0 "B [k,n](x);" "6#-&%\"BG6$%\"kG%\"nG6#%\"xG" }{TEXT -1 8 " is the " } {TEXT 275 1 "k" }{TEXT -1 4 " th " }{TEXT 260 26 "Bernstein basis poly nomial" }{TEXT -1 11 " of degree " }{TEXT 276 1 "n" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "In this \+ worksheet a reference to a Bernstein polynomial, where no underlying f unction is presumed, will mean that a basis polynomial is to be consid ered. " }}{PARA 0 "" 0 "" {TEXT -1 15 "The polynomial " }{XPPEDIT 18 0 "B[k,n](x);" "6#-&%\"BG6$%\"kG%\"nG6#%\"xG" }{TEXT -1 56 " is the Be rnstein polynomial associated with a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " which has the value 1 at " }{XPPEDIT 18 0 "x=k/n" "6#/%\"xG*&%\"kG\"\"\"%\"nG!\"\"" }{TEXT -1 15 ", and zer o for " }{XPPEDIT 18 0 "x=i/n" "6#/%\"xG*&%\"iG\"\"\"%\"nG!\"\"" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "i<>k" "6#0%\"iG%\"kG" }{TEXT -1 50 ". It is possible to define a continuous function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 22 " with this property. " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "We are i nterested in these polynomials for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!% \"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "For example, the degree 5 Bernstein polynomials a re: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([ B[0,5](x) = (1-x)^5, ``],[B[1,5](x) = 5*(1-x)^4*x, ``],[B[2,5](x) = 10 *(1-x)^3*x^2, ``],[B[3,5](x) = 10*(1-x)^2*x^3, ``],[B[4,5](x) = 5*(1-x )*x^4, ``],[B[5,5](x) = x^5, ``]);" "6#-%*PIECEWISEG6(7$/-&%\"BG6$\"\" !\"\"&6#%\"xG*$,&\"\"\"F2F/!\"\"F-%!G7$/-&F*6$F2F-6#F/*(F-F2*$,&F2F2F/ F3\"\"%F2F/F2F47$/-&F*6$\"\"#F-6#F/*(\"#5F2*$,&F2F2F/F3\"\"$F2F/FDF47$ /-&F*6$FJF-6#F/*(FGF2*$,&F2F2F/F3FDF2F/FJF47$/-&F*6$F>F-6#F/*(F-F2,&F2 F2F/F3F2F/F>F47$/-&F*6$F-F-6#F/*$F/F-F4" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "alias(C=binomial):\nn := 5;\nB := (k,n,x) \+ -> C(n,k)*(1-x)^(n-k)*x^k; \nplot([seq(B(k,n,x),k=0..n)],x=0..1,thickn ess=2,\n color=[seq(COLOR(HUE,k/(n+1)),k=0..n)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"B Gf*6%%\"kG%\"nG%\"xG6\"6$%)operatorG%&arrowGF**(-%\"CG6$9%9$\"\"\"),&F 4F49&!\"\",&F2F4F3F8F4)F7F3F4F*F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F)7$$\"3ILLL3x&) *3\"!#>$\"3A#esdH?oY*!#=7$$\"3emmm;arz@F/$\"3;(*)zY]Hm&*)F27$$\"3.++D \"y%*z7$F/$\"3VAa?]J$3`)F27$$\"3[LL$e9ui2%F/$\"3l`y6:oQ@\")F27$$\"3nmm m\"z_\"4iF/$\"35MHBU[vdsF27$$\"3[mmmT&phN)F/$\"3a]:Fd0AkkF27$$\"3CLLe* =)H\\5F2$\"3')Qe43$R\\u&F27$$\"3gmm\"z/3uC\"F2$\"3@$*[5J1pO^F27$$\"3%) ***\\7LRDX\"F2$\"3?!)G%\\-=Bc%F27$$\"3]mm\"zR'ok;F2$\"33dscDY:F27$$\"3?LLL347TLF2$\"3]gx=W* 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\"3)*\\d1C'*Fcu7$Fbr$\"3#o(\\WFJTb7F/7$Fgr$ \"31T**ydn(of\"F/7$F\\s$\"34%*pa\"fN&G?F/7$Fas$\"3\"e29PgMF]#F/7$Ffs$ \"3*[(oI\\dg4JF/7$F[t$\"37&=l=A'=fQF/7$F`t$\"3vlq0%fiMi%F/7$Fet$\"3/%= mFJ.-e&F/7$Fjt$\"3%o:aVle'GnF/7$F_u$\"3)GTkB`Rz-)F/7$Feu$\"3a9j+7#>*o% *F/7$Fju$\"3i`J&Qw*QI6F27$F_v$\"3Wr-/#*Gc=8F27$Fdv$\"3yueJq)=ja\"F27$F iv$\"3_.)HLa3*ybr\\;FF27$F^x$\"3#\\zyp\\W*)4$F27$Fcx$\"3#)>[ZTGHWNF27$Fhx$\"3EsF27$Fiz$\"33CF&)*3*Qr!) F27$Fe^n$\"3*3Gb%z'>j])F27$F^[l$\"31llbn0!)f*)F27$Fb_n$\"3)o'GgevZo%*F 27$F*F*-Fe[l6$Fg[l#\"\"&F^il-%*THICKNESSG6#F^`n-%+AXESLABELSG6$Q\"x6\" Q!6\"-%%VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k<>n" "6#0%\"kG%\"nG" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ (1-x)^(n-k)*x^k] = -(n-k)*(1-x)^(n-k-1)*x^k+(1 -x)^(n-k)*k*x^(k-1)" "6#/7#*&),&\"\"\"F(%\"xG!\"\",&%\"nGF(%\"kGF*F()F )F-F(,&*(,&F,F(F-F*F(),&F(F(F)F*,(F,F(F-F*F(F*F()F)F-F(F**(),&F(F(F)F* ,&F,F(F-F*F(F-F()F),&F-F(F(F*F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)^(n-k-1)*x^(k-1)*[k*(1-x)-( n-k)*x];" "6#/%!G*(),&\"\"\"F(%\"xG!\"\",(%\"nGF(%\"kGF*F(F*F()F),&F-F (F(F*F(7#,&*&F-F(,&F(F(F)F*F(F(*&,&F,F(F-F*F(F)F(F*F(" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)^(n-k-1 )*x^(k-1)*(k-n*x);" "6#/%!G*(),&\"\"\"F(%\"xG!\"\",(%\"nGF(%\"kGF*F(F* F()F),&F-F(F(F*F(,&F-F(*&F,F(F)F(F*F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "It follows that, for \+ " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k<>n" "6#0%\"kG%\"nG" }{TEXT -1 21 ", the derivative of " } {XPPEDIT 18 0 "B(k,n,x);" "6#-%\"BG6%%\"kG%\"nG%\"xG" }{TEXT -1 14 " i s zero when " }{XPPEDIT 18 0 "x=k/n" "6#/%\"xG*&%\"kG\"\"\"%\"nG!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Moreover the derivat ive is positive for " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 188 "alias(C=binomial):\nn := 5; \ndB := (k,n,x) -> C(n,k)*(1-x)^(n-k-1)*x^(k-1)*(k-n*x); \nplot([seq(d B(k,n,x),k=1..n-1)],x=0..1,y=-3.5..3.5,thickness=2,\ncolor=[seq(COLOR( HUE,k/(n+1)),k=1..n-1)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\" \"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dBGf*6%%\"kG%\"nG%\"xG6\"6$% )operatorG%&arrowGF***-%\"CG6$9%9$\"\"\"),&F4F49&!\"\",(F2F4F3F8F4F8F4 )F7,&F3F4F4F8F4,&F3F4*&F2F4F7F4F8F4F*F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 410 311 311 {PLOTDATA 2 "6)-%'CURVESG6$7X7$$\"\"!F)$\"\"&F)7 $$\"3ILLL3x&)*3\"!#>$\"3Wue/;QkuX!#<7$$\"3emmm;arz@F/$\"3-SuP;90qTF27$ $\"3.++D\"y%*z7$F/$\"3A[$\\5vOW$QF27$$\"3[LL$e9ui2%F/$\"3;KUd\\\\o8NF2 7$$\"33++voMrU^F/$\"3%3ZwD/O-<$F27$$\"3nmmm\"z_\"4iF/$\"3>8hty=aWGF27$ $\"3emmmm6m#G(F/$\"3?kRr@I2MDF27$$\"3[mmmT&phN)F/$\"3$[![%oE.0C#F27$$ \"36++v=ddC%*F/$\"3!QRP$ykdk>F27$$\"3CLLe*=)H\\5!#=$\"32%[Tw[PVq\"F27$ $\"3gmm\"z/3uC\"Fhn$\"3305Iz\\ch7F27$$\"3%)***\\7LRDX\"Fhn$\"3uC/A]U\" oa)Fhn7$$\"3]mm\"zR'ok;Fhn$\"3e'))f%)zRY&[Fhn7$$\"3w***\\i5`h(=Fhn$\"3 Cm]P=&3+m\"Fhn7$$\"3WLLL3En$4#Fhn$!3IF9,ZFQd6Fhn7$$\"3qmm;/RE&G#Fhn$!3 [JM!G9QXF$Fhn7$$\"3\")*****\\K]4]#Fhn$!3CYRj6BX\"G&Fhn7$$\"3$******\\P Avr#Fhn$!3L:CIS13GpFhn7$$\"3)******\\nHi#HFhn$!3[C67W9='>)Fhn7$$\"3jmm \"z*ev:JFhn$!3I=joWnw+\"*Fhn7$$\"3?LLL347TLFhn$!37#=]H+c%**)*Fhn7$$\"3 ,LLLLY.KNFhn$!3r%QRCSij.\"F27$$\"3w***\\7o7Tv$Fhn$!3)\\6J&)H8&o5F27$$ \"3'GLLLQ*o]RFhn$!3C\"z]D^d&z5F27$$\"3A++D\"=lj;%Fhn$!3;LAsR.?v5F27$$ \"31++vV&RH'Qf-\"F27$$ \"3GLLeR\"3Gy%Fhn$!31Hi0g6Zz)*Fhn7$$\"3cmm;/T1&*\\Fhn$!3Sm;>J9K(Q*Fhn7 $$\"3&em;zRQb@&Fhn$!3K*[`fThU!))Fhn7$$\"3\\***\\(=>Y2aFhn$!31NoAo=V^#) Fhn7$$\"39mm;zXu9cFhn$!3=YQUl0$3i(Fhn7$$\"3l******\\y))GeFhn$!3_!yZoVN l%pFhn7$$\"3'*)***\\i_QQgFhn$!3EIAJ<%yrF'Fhn7$$\"3@***\\7y%3TiFhn$!3a$o'4/u#\\Fhn7$$\"3kKLL$Qx$omFhn $!3qB?QS)>fJ%Fhn7$$\"3!)*****\\P+V)oFhn$!3c1K()3ND$p$Fhn7$$\"3?mm\"zpe *zqFhn$!3u,m\\T'Q?;$Fhn7$$\"3%)*****\\#\\'QH(Fhn$!3-Jxt]kxAEFhn7$$\"3G KLe9S8&\\(Fhn$!3KnX4c^4f@Fhn7$$\"3R***\\i?=bq(Fhn$!3w.!#@7$$\"\"\"F)F(-%&COLORG6$%$HUEG#Fa\\l\"\"'-F$6$7in7$F(F(7$F-$ \"3)*H8_/2Ou?Fhn7$F4$\"3>[G$\"3^2%>$ \\X+PnFhn7$FC$\"3e#*pQI(o[1)Fhn7$FH$\"3:Z]%oq8$G#*Fhn7$FM$\"3Im=\"[dTT -\"F27$FR$\"3*H1RT*GQ56F27$Ffn$\"3-)y&G'pZ-C\"F27$$\"3')***\\(=JN[6Fhn $\"39rj5Q)*)GG\"F27$F\\o$\"3YZV3#*e?:8F27$$\"33LLe*ot*\\8Fhn$\"3p$=5V \"GQQ8F27$Fao$\"3_'f%[Cyp^8F27$$\"3emm\"z4wb]\"Fhn$\"3!Q[fv?6\\N\"F27$ $\"3/LLekGhe:Fhn$\"3]GH%y%zsb8F27$$\"3x***\\7j\\;h\"Fhn$\"3]s4xJ!GUN\" F27$Ffo$\"3#)>=H'\\!\\]8F27$$\"39LL3_(>/x\"Fhn$\"3:k!H:irmL\"F27$F[p$ \"35bY<[k([J\"F27$F`p$\"35q)ezwduC\"F27$Fep$\"3GOwhro7m6F27$Fjp$\"3oAI \"Hx_T0\"F27$F_q$\"3P,_!3$ooT#*Fhn7$Fdq$\"3QnX(=1_7'yFhn7$Fiq$\"3$ztA* *)3dGlFhn7$F^r$\"3u+!f/!fc!)[Fhn7$Fcr$\"3WI;5MbNdMFhn7$Fhr$\"3jR.upc`+ =Fhn7$F]s$\"3qD8(H1rWc$F/7$Fbs$!3!o=DAO>%z6Fhn7$Fgs$!31Da&=0?h-)Fhn7$Feu$!3u!f%fYO`<()Fhn7$Fju$!3WV%* )=cgNF*Fhn7$F_v$!3911ZdVwe'*Fhn7$Fdv$!3wO\"[%)=&G\"))*Fhn7$Fiv$!3+\"e4 o]oq&**Fhn7$F^w$!3c&yFa8r_()*Fhn7$Fcw$!3[0_e>*yyj*Fhn7$Fhw$!3)f,Z*Q&)e 'H*Fhn7$F]x$!35&3i;bcpz)Fhn7$Fbx$!3(H^\\B`/$=#)Fhn7$Fgx$!3OE(yRacg^(Fh n7$F\\y$!3_R!Hhvj,v'Fhn7$Fay$!3'eGw[:P])eFhn7$Ffy$!3zVO*)G#[G,&Fhn7$F[ z$!3H'zlM;Zl5%Fhn7$F`z$!3X0`+(zC2A$Fhn7$Fez$!3%o\"*R]CfNW#Fhn7$Fjz$!3n ?,Mww+H;Fhn7$F_[l$!3Y:F'36*e+5Fhn7$Fe[l$!3=G=zWHh.ZF/7$$\"3C+++]oi\"o* Fhn$!3'>C&32[\"yy#F/7$Fj[l$!3)fJAR(oNN8F/7$$\"35+]P40O\"*)*Fhn$!3KNmEJ /))QMFc[lF_\\l-Fc\\l6$Fe\\l#Fa\\l\"\"$-F$6$7inF[]l7$F-$\"37#)G*Q980Y$F c[l7$F4$\"3l&z%ojqiV8F/7$F9$\"3))GFI#))[_p#F/7$F>$\"3aP(G!RlvcWF/7$FH$ \"3r8#)Gm>JD(*F/7$FR$\"3I2$GfplBl\"Fhn7$Ffn$\"3u2)4,\")[%RCFhn7$F\\o$ \"3Fz!['p6MOKFhn7$Fao$\"3R8=N'G`/5%Fhn7$Ffo$\"30![.3(p*p+&Fhn7$F[p$\"3 QR8.%=\"='*eFhn7$F`p$\"3ewm#))Fhn7$Fdq$\"3$)H\\/(4K\"4$*Fhn7$Fiq$ \"3%>)z1R-'zj*Fhn7$F^r$\"334b&Rk5A))*Fhn7$Fcr$\"3=9&\\/gdp&**Fhn7$Fhr$ \"3'y\"yq^Mx%))*Fhn7$F]s$\"3Isj$e^?Xn*Fhn7$Fbs$\"3ge?\\RZ.%G*Fhn7$Fgs$ \"3U/GpQ!Huv)Fhn7$F\\t$\"3%f!H;,t3]!)Fhn7$Fat$\"35^v%*=IEjsFhn7$Fft$\" 3%f6W0\"HkuiFhn7$F[u$\"3_e2clLt/^Fhn7$F`u$\"3S0$fDfg&yRFhn7$Feu$\"3;2Y @D`,jEFhn7$Fju$\"3UDt5,(yC@\"Fhn7$F_v$!3aIt-`@NsFF/7$Fdv$!3c3n&4s4\\w \"Fhn7$Fiv$!3q[%=W'\\wVMFhn7$F^w$!3eu\"f?gU4&\\Fhn7$Fcw$!3i`xS$=v*GlFh n7$Fhw$!3aX=;j![O!zFhn7$F]x$!35%fkFOMPJ*Fhn7$Fbx$!3q%Hv;&f%>0\"F27$Fgx $!3I\\)3BNb<;\"F27$F\\y$!3-_iV_nF\\7F27$Fay$!3Y+^Y)zEbJ\"F27$$\"3nK$e* [ACI#)Fhn$!3Z!pz^')Fhn$!3*\\o'e0Y]of2 C\"F27$Fjz$!3vVg6$))o`5\"F27$$\"3y***\\(oTAq#*Fhn$!3\")>?/aqXD5F27$F_[ l$!3cYMGMkAF$*Fhn7$$\"3sK$eRA5\\Z*Fhn$!3kX0MaQD!>)Fhn7$Fe[l$!3SBr]w1z# *oFhn7$Fahl$!3ywdq]5U$\\&Fhn7$Fj[l$!3a!*HE8h)G$RFhn7$Fihl$!3kcWPj')4o? FhnF_\\l-Fc\\l6$Fe\\l#Fa\\l\"\"#-F$6$7WF[]l7$F4$\"38\\![pC,[,#F^\\l7$F >$\"3s\"p\"4f`g&G\"Fc[l7$FH$\"3!e/65Fhn7$F`p$\"3%*4I/$)*Q^N\"Fhn7$Fep$\"3' o;*R%*>30#)*Q%o^]@Fhn7$F_q$\"3'fd(y^BJ]EFhn7$Fdq$\"3qw= ub%3$yJFhn7$Fiq$\"3kwN.s5T$p$Fhn7$F^r$\"3B)3&*H5#3WVFhn7$Fcr$\"3Z.QB*) z!=#\\Fhn7$Fhr$\"3Z>*Q1\\Ugh&Fhn7$F]s$\"3OEV(*e)HAC'Fhn7$Fbs$\"3h7?#)= &G9$pFhn7$Fgs$\"3]8 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Note that, for " } {XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\" \"\"" }{TEXT -1 82 ", the values of all the Bernstein polynomials of a given degree sum to 1, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(B[k,n](x),k = 0 .. n) = 1;" "6#/-%$SumG6$-&%\"BG6$% \"kG%\"nG6#%\"xG/F+;\"\"!F,\"\"\"" }{TEXT -1 14 " ------- (i). " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 9 "_________" }{TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 63 "We say that the Bernstein polynomials of a given degree form a " }{TEXT 260 18 "partition of unity" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The formula (i) is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(matrix([[ n], [k]])*(1-x)^(n-k)*x^k,k = 0 .. n) = 1;" "6#/-%$SumG6$*(-%'matrixG6 #7$7#%\"nG7#%\"kG\"\"\"),&F0F0%\"xG!\"\",&F-F0F/F4F0)F3F/F0/F/;\"\"!F- F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 41 "which follows from the binomial theorem: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(p+q)^n = Sum(matrix([[n], [k]])*p^(n-k)*q^k,k = 0 .. n);" "6#/) ,&%\"pG\"\"\"%\"qGF'%\"nG-%$SumG6$*(-%'matrixG6#7$7#F)7#%\"kGF')F&,&F) F'F4!\"\"F')F(F4F'/F4;\"\"!F)" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "by taking " }{XPPEDIT 18 0 "p = 1-x;" "6#/%\"pG,&\"\"\"F&%\"xG!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "q = x;" "6#/%\"qG%\"xG" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "p+q=1" "6#/,&%\"pG\"\"\"%\"qGF&F&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "The form ula (i) can also be interpreted in the context of a probability experi ment involving repeated trials with a probability " }{TEXT 267 1 "p" } {TEXT -1 31 " of success, and a probability " }{TEXT 266 1 "q" }{TEXT -1 21 " of failure (so that " }{XPPEDIT 18 0 "p+q=1" "6#/,&%\"pG\"\"\" %\"qGF&F&" }{TEXT -1 21 "). The probability of" }{XPPEDIT 18 0 "``(n-k );" "6#-%!G6#,&%\"nG\"\"\"%\"kG!\"\"" }{TEXT -1 16 " successes (and " }{TEXT 264 1 "k" }{TEXT -1 32 " failures), in any order, after " } {TEXT 265 1 "n" }{TEXT -1 11 " trials is " }{XPPEDIT 18 0 "matrix([[n] , [k]])*p^(n-k)*q^k" "6#*(-%'matrixG6#7$7#%\"nG7#%\"kG\"\"\")%\"pG,&F) F,F+!\"\"F,)%\"qGF+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 " Hence the probability of somewhere between 0 and " }{TEXT 277 1 "n" } {TEXT -1 42 " successes (which is certain to occur) is " }{XPPEDIT 18 0 "Sum(matrix([[n], [k]])*p^(n-k)*q^k,k = 0 .. n)=1" "6#/-%$SumG6$*(-% 'matrixG6#7$7#%\"nG7#%\"kG\"\"\")%\"pG,&F-F0F/!\"\"F0)%\"qGF/F0/F/;\" \"!F-F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "A recursive definition of the Bernstein polynomials " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "The Bernstein polynomials of degree " }{TEXT 278 1 "n" }{TEXT -1 75 " can be defined by \"blending together\" two Bernstein polynomi als of degree " }{XPPEDIT 18 0 "n-1" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "More precisely, the " }{TEXT 279 1 "k" }{TEXT -1 11 " th degree " }{TEXT 280 1 "n" }{TEXT -1 40 " B ernstein polynomial can be written as " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B[k,n](x) = (1-x) *B[k,n-1](x)+x*B[k-1,n-1](x);" "6#/-&%\"BG6$%\"kG%\"nG6#%\"xG,&*&,&\" \"\"F/F+!\"\"F/-&F&6$F(,&F)F/F/F06#F+F/F/*&F+F/-&F&6$,&F(F/F/F0,&F)F/F /F06#F+F/F/" }{TEXT -1 13 " ------- (i)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 28 "____________________________" }{TEXT -1 17 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "This formula is a consequence of the formula connecting t he binomial coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[n-1], [k]])+matrix([[n-1], [k-1]]) = matrix([[ n], [k]]);" "6#/,&-%'matrixG6#7$7#,&%\"nG\"\"\"F,!\"\"7#%\"kGF,-F&6#7$ 7#,&F+F,F,F-7#,&F/F,F,F-F,-F&6#7$7#F+7#F/" }{TEXT -1 14 " ------- (ii) ," }}{PARA 0 "" 0 "" {TEXT -1 77 "which coresponds to the standard met hod of construction of Pascal's triangle." }}{PARA 0 "" 0 "" {TEXT -1 15 "Proof of (ii): " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[n-1], [k]])+matrix([[n-1], [k-1]]) = (n-1)!/(k!*(n-k-1)!) \+ + (n-1)!/((k-1)!*(n-k)!)" "6#/,&-%'matrixG6#7$7#,&%\"nG\"\"\"F,!\"\"7# %\"kGF,-F&6#7$7#,&F+F,F,F-7#,&F/F,F,F-F,,&*&-%*factorialG6#,&F+F,F,F-F ,*&-F:6#F/F,-F:6#,(F+F,F/F-F,F-F,F-F,*&-F:6#,&F+F,F,F-F,*&-F:6#,&F/F,F ,F-F,-F:6#,&F+F,F/F-F,F-F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ((n-k) *(n-1)!+k*(n-1)!)/(k!*(n-k)!);" "6#/%!G*&,&*&,&%\"nG\"\"\"%\"kG!\"\"F* -%*factorialG6#,&F)F*F*F,F*F**&F+F*-F.6#,&F)F*F*F,F*F*F**&-F.6#F+F*-F. 6#,&F)F*F+F,F*F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = n!/(k!*(n-k)!);" "6#/%!G*&-%*factorialG6#%\"nG\"\" \"*&-F'6#%\"kGF*-F'6#,&F)F*F.!\"\"F*F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = matrix([[n], [k]]);" "6#/%!G-%'matrixG6#7$7#%\"nG7#%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Proof of (i):" }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(1-x)*B[k,n-1](x)+x*B[k-1,n-1]( x);" "6#,&*&,&\"\"\"F&%\"xG!\"\"F&-&%\"BG6$%\"kG,&%\"nGF&F&F(6#F'F&F&* &F'F&-&F+6$,&F-F&F&F(,&F/F&F&F(6#F'F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = (1-x)*matrix([[n-1], [k]])*(1-x)^(n-1-k)*x^k*` `+` `*x*matrix([ [n-1], [k-1]])*(1-x)^(n-k)*x^(k-1);" "6#/%!G,&*,,&\"\"\"F(%\"xG!\"\"F( -%'matrixG6#7$7#,&%\"nGF(F(F*7#%\"kGF(),&F(F(F)F*,(F1F(F(F*F3F*F()F)F3 F(%#~~GF(F(*,F8F(F)F(-F,6#7$7#,&F1F(F(F*7#,&F3F(F(F*F(),&F(F(F)F*,&F1F (F3F*F()F),&F3F(F(F*F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (` `* matrix([[n-1], [k]])+matrix([[n-1], [k]])*` `)*(1-x)^(n-k)*x^k;" "6#/% !G*(,&*&%\"~G\"\"\"-%'matrixG6#7$7#,&%\"nGF)F)!\"\"7#%\"kGF)F)*&-F+6#7 $7#,&F0F)F)F17#F3F)F(F)F)F)),&F)F)%\"xGF1,&F0F)F3F1F))F=F3F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = matri x([[n], [k]])*(1-x)^(n-k)*x^k;" "6#/%!G*(-%'matrixG6#7$7#%\"nG7#%\"kG \"\"\"),&F.F.%\"xG!\"\",&F+F.F-F2F.)F1F-F." }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 29 ": The formula (ii) holds fo r " }{XPPEDIT 18 0 "k<0" "6#2%\"kG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "k>n" "6#2%\"nG%\"kG" }{TEXT -1 44 " provided that we ad opt the convention that " }{XPPEDIT 18 0 "matrix([[n], [k]])=0" "6#/-% 'matrixG6#7$7#%\"nG7#%\"kG\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k <0" "6#2%\"kG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k>n" "6#2%\"nG %\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "To be consistent with this convention, we define " }{XPPEDIT 18 0 "B[k,n](x);" "6#-&% \"BG6$%\"kG%\"nG6#%\"xG" }{TEXT -1 28 " to be identically zero for " } {XPPEDIT 18 0 "k<0" "6#2%\"kG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k>n" "6#2%\"nG%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 75 "An inductive proof that the Bernstein polynomials \+ form a partition of unity" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 21 "The recursive formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B[k,n](x) = (1-x)*B[k,n-1](x)+x *B[k-1,n-1](x);" "6#/-&%\"BG6$%\"kG%\"nG6#%\"xG,&*&,&\"\"\"F/F+!\"\"F/ -&F&6$F(,&F)F/F/F06#F+F/F/*&F+F/-&F&6$,&F(F/F/F0,&F)F/F/F06#F+F/F/" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "which enables Bernstein polynomials of a given degree " } {TEXT 281 1 "n" }{TEXT -1 49 " to be constructed from the polynomials \+ of degree" }{XPPEDIT 18 0 "``(n-1);" "6#-%!G6#,&%\"nG\"\"\"F(!\"\"" } {TEXT -1 80 ", provides an alternative way of seeing that the Bernstei n polynomials sum to 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Starting with the obvious fact that the two degree1 \+ Bernstein polynomials: " }{XPPEDIT 18 0 "B[0,1](x) = 1-x;" "6#/-&%\"BG 6$\"\"!\"\"\"6#%\"xG,&F)F)F+!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[1,1](x) = x;" "6#/-&%\"BG6$\"\"\"F(6#%\"xGF*" }{TEXT -1 80 ", sum t o 1, we can consider the sum of the three degree 2 Bernstein polynomia ls:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "B[0,2](x)+B[1,2](x)+B[2,2](x);" "6#,(-&%\"BG6$\"\"! \"\"#6#%\"xG\"\"\"-&F&6$F,F)6#F+F,-&F&6$F)F)6#F+F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (1-x)*B[0,1](x)+x*B[-1,1](x)+(1-x)*B[1,1](x)+x*B[0 ,1](x)+(1-x)*B[2,1](x)+x*B[1,1](x);" "6#/%!G,.*&,&\"\"\"F(%\"xG!\"\"F( -&%\"BG6$\"\"!F(6#F)F(F(*&F)F(-&F-6$,$F(F*F(6#F)F(F(*&,&F(F(F)F*F(-&F- 6$F(F(6#F)F(F(*&F)F(-&F-6$F/F(6#F)F(F(*&,&F(F(F)F*F(-&F-6$\"\"#F(6#F)F (F(*&F)F(-&F-6$F(F(6#F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)* [B[0,1](x)+B[1,1](x)+B[2,1](x)]+x*[B[-1,1](x)+B[0,1](x)+B[1,1](x)];" " 6#/%!G,&*&,&\"\"\"F(%\"xG!\"\"F(7#,(-&%\"BG6$\"\"!F(6#F)F(-&F/6$F(F(6# F)F(-&F/6$\"\"#F(6#F)F(F(F(*&F)F(7#,(-&F/6$,$F(F*F(6#F)F(-&F/6$F1F(6#F )F(-&F/6$F(F(6#F)F(F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)*`.`*1 +x*`.`*1;" "6#/%!G,&*(,&\"\"\"F(%\"xG!\"\"F(%\".GF(F(F(F(*(F)F(F+F(F(F (F(" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "In a similar way, \+ we can now show that the sum of the four degree 3 Bernstein polynomial s sum to 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B[0,3](x)+B[1,3](x)+B[2,3](x)+B[3,3](x) ;" "6#,*-&%\"BG6$\"\"!\"\"$6#%\"xG\"\"\"-&F&6$F,F)6#F+F,-&F&6$\"\"#F)6 #F+F,-&F&6$F)F)6#F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)*B[0,2] (x)+x*B[-1,2](x)+(1-x)*B[1,2](x)+x*B[0,2](x)+(1-x)*B[2,2](x)+x*B[1,2]( x)+(1-x)*B[3,2](x)+x*B[2,2](x);" "6#/%!G,2*&,&\"\"\"F(%\"xG!\"\"F(-&% \"BG6$\"\"!\"\"#6#F)F(F(*&F)F(-&F-6$,$F(F*F06#F)F(F(*&,&F(F(F)F*F(-&F- 6$F(F06#F)F(F(*&F)F(-&F-6$F/F06#F)F(F(*&,&F(F(F)F*F(-&F-6$F0F06#F)F(F( *&F)F(-&F-6$F(F06#F)F(F(*&,&F(F(F)F*F(-&F-6$\"\"$F06#F)F(F(*&F)F(-&F-6 $F0F06#F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)*[B[0,2](x) +B[1,2](x)+B[2,2](x)+B[3,2](x)]+x*[B[-1,2](x)+B[0,2](x)+B[1,2](x)+B[2, 2](x)];" "6#/%!G,&*&,&\"\"\"F(%\"xG!\"\"F(7#,*-&%\"BG6$\"\"!\"\"#6#F)F (-&F/6$F(F26#F)F(-&F/6$F2F26#F)F(-&F/6$\"\"$F26#F)F(F(F(*&F)F(7#,*-&F/ 6$,$F(F*F26#F)F(-&F/6$F1F26#F)F(-&F/6$F(F26#F)F(-&F/6$F2F26#F)F(F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)*`.`*1+x*`.`*1;" "6#/%!G,&*(, &\"\"\"F(%\"xG!\"\"F(%\".GF(F(F(F(*(F)F(F+F(F(F(F(" }{XPPEDIT 18 0 " ` `= 1" "6#/%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 57 "In general, if we have reached the stage of showing that " }{XPPEDIT 18 0 "Sum(B[k,n-1](x),k = 0 .. n-1) = 1; " "6#/-%$SumG6$-&%\"BG6$%\"kG,&%\"nG\"\"\"F.!\"\"6#%\"xG/F+;\"\"!,&F-F .F.F/F." }{TEXT -1 19 ", we can show that " }{XPPEDIT 18 0 "Sum(B[k,n] (x),k = 0 .. n) = 1;" "6#/-%$SumG6$-&%\"BG6$%\"kG%\"nG6#%\"xG/F+;\"\"! F,\"\"\"" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(B[k,n](x),k = 0 .. n) = Sum((1-x)*B[k,n-1](x)+x*B[k-1,n-1](x),k = 0 .. n);" "6#/-%$Su mG6$-&%\"BG6$%\"kG%\"nG6#%\"xG/F+;\"\"!F,-F%6$,&*&,&\"\"\"F7F.!\"\"F7- &F)6$F+,&F,F7F7F86#F.F7F7*&F.F7-&F)6$,&F+F7F7F8,&F,F7F7F86#F.F7F7/F+;F 1F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)*Sum(B[k,n-1](x),k = 0 .. n)+x*Sum(B[k-1,n-1](x),k = 0 .. n);" "6#/%!G,&*&,&\"\"\"F(%\"xG!\"\"F(-%$SumG6$-&%\"BG6$%\"kG,&%\"nGF (F(F*6#F)/F2;\"\"!F4F(F(*&F)F(-F,6$-&F06$,&F2F(F(F*,&F4F(F(F*6#F)/F2;F 8F4F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x)*`.`*1+x*`.`*1;" "6#/% !G,&*(,&\"\"\"F(%\"xG!\"\"F(%\".GF(F(F(F(*(F)F(F+F(F(F(F(" }{XPPEDIT 18 0 "``= 1" "6#/%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "De Casteljau's algorithm " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The Bernstein basis polynomials " } {XPPEDIT 18 0 "B[k,n](x) = matrix([[n], [k]])*(1-x)^(n-k)*x^k;" "6#/-& %\"BG6$%\"kG%\"nG6#%\"xG*(-%'matrixG6#7$7#F)7#F(\"\"\"),&F3F3F+!\"\",& F)F3F(F6F3)F+F(F3" }{TEXT -1 9 ", where " }{XPPEDIT 18 0 "matrix([[n] , [k]]);" "6#-%'matrixG6#7$7#%\"nG7#%\"kG" }{TEXT -1 29 " is the binom ial coefficient " }{XPPEDIT 18 0 "n!/(k!*(n-k)!)" "6#*&-%*factorialG6# %\"nG\"\"\"*&-F%6#%\"kGF(-F%6#,&F'F(F,!\"\"F(F0" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 12 "satisfy the " }{TEXT 260 19 "recurrence r elation" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B[k,n](x) = (1-x)*B[k,n-1](x)+x* B[k-1,n-1](x);" "6#/-&%\"BG6$%\"kG%\"nG6#%\"xG,&*&,&\"\"\"F/F+!\"\"F/- &F&6$F(,&F)F/F/F06#F+F/F/*&F+F/-&F&6$,&F(F/F/F0,&F)F/F/F06#F+F/F/" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 27 "____ _______________________" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "This relation can be exploited \+ to provide an algorithm for evaluating a polynomial " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = c[0]*B[0,n](x)+c[1]*B[1,n]( x)+` . . . `+c[n]*B[n,n](x);" "6#/-%\"pG6#%\"xG,**&&%\"cG6#\"\"!\"\"\" -&%\"BG6$F-%\"nG6#F'F.F.*&&F+6#F.F.-&F16$F.F36#F'F.F.%(~.~.~.~GF.*&&F+ 6#F3F.-&F16$F3F36#F'F.F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "given in terms of the Bernstein co efficients " }{XPPEDIT 18 0 "c[0],c[1],` . . . `,c[n]" "6&&%\"cG6#\"\" !&F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 33 " relative to the Berns tein basis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 24 "De Casteljau's algorithm" }{TEXT -1 31 " (implemented below) d oes this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "deCasteljau := proc(c::list,x::algebraic)\n local \+ i,j,p,n;\n n := nops(c)-1;\n p := array(0..n,0..n);\n\n for i fr om 0 to n do p[i,n] := c[i+1] end do;\n for j from 1 to n do\n \+ for i from 0 to n-j do \n p[i,n-j] := (1-x)*p[i,n-j+1]+x*p[i+1 ,n-j+1];\n end do;\n end do;\n p[0,0];\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Example:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x/(1+ x)" "6#/-%\"fG6#%\"xG*&F'\"\"\",&F)F)F'F)!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 25 "For any positive integer " }{TEXT 282 1 " n" }{TEXT -1 26 " the Bernstein polynomial " }{XPPEDIT 18 0 "B(n,x)" " 6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 17 " associated with " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x)=Sum(f(k/n)*B[k,n](x),k = 0 .. n)" "6#/ -%\"BG6$%\"nG%\"xG-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"F'!\"\"F2-&F%6$F1F'6 #F(F2/F1;\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "We ca n use the de Casteljau algorithm to evaluate " }{XPPEDIT 18 0 "B(n,x) " "6#-%\"BG6$%\"nG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "n := 5;\nf := x -> x/( 1+x);\nxvals := [seq(k/n,k=0..n)];\nyvals := map(f,xvals);\ndeCastelja u(yvals,0.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9 $\"\"\",&F.F.F-F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xva lsG7(\"\"!#\"\"\"\"\"&#\"\"#F)#\"\"$F)#\"\"%F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7(\"\"!#\"\"\"\"\"'#\"\"#\"\"(#\"\"$\"\")#\"\" %\"\"*#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l;*Q-%!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "The proce dure can be used to draw a graph of the associated Bernstein polynomia l." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(deCasteljau(yvals,x),x=0..1);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\" 3emmm;arz@!#>$\"3O$R#o$)*))Rz\"F-7$$\"3[LL$e9ui2%F-$\"3=sJCQ)p*=LF-7$$ \"3nmmm\"z_\"4iF-$\"3s,5pS;#\\*\\F-7$$\"3[mmmT&phN)F-$\"3!*RpdI#*>TmF- 7$$\"3CLLe*=)H\\5!#=$\"3_#)y***f2+C)F-7$$\"3gmm\"z/3uC\"FB$\"3wZ3<5Q\" yo*F-7$$\"3%)***\\7LRDX\"FB$\"3Am1CgzF:6FB7$$\"3]mm\"zR'ok;FB$\"3S^@$* pd@j7FB7$$\"3w***\\i5`h(=FB$\"3vu[m%earS\"FB7$$\"3WLLL3En$4#FB$\"3A_!o ToK;b\"FB7$$\"3qmm;/RE&G#FB$\"3AK0MwO&fn\"FB7$$\"3\")*****\\K]4]#FB$\" 3K9Pw(*)*p7=FB7$$\"3$******\\PAvr#FB$\"3gAB$HOqm%>FB7$$\"3)******\\nHi #HFB$\"3qC'Q.=\"ps?FB7$$\"3jmm\"z*ev:JFB$\"3e.qZ73d%=#FB7$$\"3?LLL347T LFB$\"3FVt!Q:9XJ#FB7$$\"3,LLLLY.KNFB$\"3T*R$*R5G?U#FB7$$\"3w***\\7o7Tv $FB$\"3g!o,l[2Ua#FB7$$\"3'GLLLQ*o]RFB$\"39,m(*3:$)\\EFB7$$\"3A++D\"=lj ;%FB$\"3#*yM[N*eIw#FB7$$\"31++vV&R2$FB7$$\"3cmm;/T1&*\\ FB$\"3-/,Y\\0^tJFB7$$\"3&em;zRQb@&FB$\"3e=Cog#okF$FB7$$\"3\\***\\(=>Y2 aFB$\"37KM)39lSO$FB7$$\"39mm;zXu9cFB$\"3#Q$o$3#fgcMFB7$$\"3l******\\y) )GeFB$\"3'QK9&4K,]NFB7$$\"3'*)***\\i_QQgFB$\"33-#QQR!HROFB7$$\"3@***\\ 7y%3TiFB$\"3![Pd#opW^[)o?%FB7$$\"3R***\\i?=bq(FB$\"3u%o+oUer?(=j%FB7$$\"3u******\\Qk\\*)FB$\"3_\"o`@4(> #p%FB7$$\"3CLL$3dg6<*FB$\"3kHX)e1x)fZFB7$$\"3ImmmmxGp$*FB$\"3q951kY7>[ FB7$$\"3A++D\"oK0e*FB$\"3'=dv\"fH*4)[FB7$$\"3A++v=5s#y*FB$\"3(R;\"f!)G )*Q\\FB7$$\"\"\"F)$\"3++++++++]FB-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AX ESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "A procedure implementing de Casteljau's algorithm: " }{TEXT 0 9 "casteljau" }{TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "ca steljau: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 272 4 " " }{TEXT -1 18 "casteljau(pts, x) " }{TEXT 274 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parame ters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " pts - a list of values " }{XPPEDIT 18 0 "[y[1],y[2],` . \+ . . `,y[n]]" "6#7&&%\"yG6#\"\"\"&F%6#\"\"#%(~.~.~.~G&F%6#%\"nG" } {TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 18 " " } {TEXT 260 2 "OR" }{TEXT -1 19 " a list of points " }{XPPEDIT 18 0 "[[ x[0], y[0]], [x[1], y[1]], ` . . . `, [x[n], y[n]]];" "6#7&7$&%\"xG6# \"\"!&%\"yG6#F(7$&F&6#\"\"\"&F*6#F/%(~.~.~.~G7$&F&6#%\"nG&F*6#F6" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 139 " x - the variable to be used in the descriptio n of the polynomial(s), or the input value at which to evaluate the po lynomial(s) " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 257 "" 0 " " {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proced ure " }{TEXT 0 9 "casteljau" }{TEXT -1 81 " uses de Casteljau's algori thm to construct or evaluate the Bernstein polynomial " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = Sum(y[k]*matrix([[n], [k]])* (1-x)^(n-k)*x^k,k = 0 .. n);" "6#/%\"yG-%$SumG6$**&F$6#%\"kG\"\"\"-%'m atrixG6#7$7#%\"nG7#F+F,),&F,F,%\"xG!\"\",&F2F,F+F7F,)F6F+F,/F+;\"\"!F2 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 97 "associated with give n data values, or to construct or evaluate the pair of Bernstein polyn omials " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = Sum(x[ k]*matrix([[n], [k]])*(1-t)^(n-k)*t^k,k = 0 .. n);" "6#/%\"xG-%$SumG6$ **&F$6#%\"kG\"\"\"-%'matrixG6#7$7#%\"nG7#F+F,),&F,F,%\"tG!\"\",&F2F,F+ F7F,)F6F+F,/F+;\"\"!F2" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = Sum(y[k]*matrix([[n], [k]])*(1-t)^(n-k)*t^k, k = 0 .. n);" "6#/%\"yG-%$SumG6$**&F$6#%\"kG\"\"\"-%'matrixG6#7$7#%\"n G7#F+F,),&F,F,%\"tG!\"\",&F2F,F+F7F,)F6F+F,/F+;\"\"!F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 273 8 "Options:" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 29 "Wi th the option \"info=true\", " }{TEXT 0 9 "casteljau" }{TEXT -1 67 " g ives the intermediate values obtained by de Casteljau's algorithm" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "plot=true /false" }}{PARA 0 "" 0 "" {TEXT -1 30 "With the option \"plot=true\", \+ " }{TEXT 0 9 "casteljau" }{TEXT -1 55 " gives output data in a form \+ which can be supplied to " }{TEXT 0 4 "plot" }{TEXT -1 28 " in order t o plot the curve." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 86 " : This option only applies in the case where the second parameter is o f type \"name\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "How to activate:" } {TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the \+ subsection, place the cursor anywhere after the prompt [ > and press \+ [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "casteljau: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3765 "casteljau := pro c(pts::list,x::algebraic)\n local pp,Options,prntflg,np,n,nmrc,flt,p nts,p,q,i,j,plt;\n\n if not (type(pts,list(algebraic)) or type(pts,l istlist(algebraic))) then\n error \"the lst argument, %1, is inva lid .. it should be either a list of values, or a list of points\",pts ;\n end if;\n \n # Get the options we need to process.\n Options := [];\n plt := false;\n prntflg := false;\n if nargs>=3 then\n Options := [args[3..nargs]];\n if not type(Options,list(equ ation)) then\n error \"each optional argument must be an equat ion\"\n end if;\n if hasoption(Options,'plot','plt','Options ') then\n if plt<>true then plt := false end if;\n end if ;\n if hasoption(Options,'info','prntflg','Options') then\n \+ if prntflg<>true then prntflg := false end if;\n end if;\n \+ if nops(Options)>0 then\n error \"%1 is not a valid option f or %2\",op(1,Options),procname;\n end if;\n end if;\n\n np := nops(pts);\n n := np-1;\n pnts := type(pts,listlist);\n\n flt : = type(x,float);\n nmrc := type(x,complexcons);\n if pnts then\n \+ for i from 1 to np do\n if not type(pts[i],list(complexcon s)) then\n nmrc := false; # not numeric\n break; \n end if;\n end do;\n if flt then pp := map(evalf,p ts)\n elif not nmrc then pp := map(normal,pts)\n else pp := \+ pts end if;\n\n if prntflg then\n p := array(0..n,0..n); \n q := array(0..n,0..n);\n\n for i from 0 to n do\n \+ p[i,n] := pp[i+1,1];\n q[i,n] := pp[i+1,2];\n \+ end do;\n for j from 0 to n-1 do\n for i from \+ 0 to n do\n p[i,j] := 0;\n q[i,j] := 0;\n \+ end do;\n end do;\n else\n p := table( );\n q := table();\n for i from 0 to n do\n \+ p[i,n] := pp[i+1,1];\n q[i,n] := pp[i+1,2];\n end \+ do;\n end if;\n\n # de Casteljau's algorithm\n for j fr om 1 to n do\n for i from 0 to n-j do \n p[i,n-j] : = (1-x)*p[i,np-j]+x*p[i+1,np-j];\n q[i,n-j] := (1-x)*q[i,np -j]+x*q[i+1,np-j];\n if not nmrc then\n p[i,n -j] := normal(p[i,n-j]);\n q[i,n-j] := normal(q[i,n-j]); \n end if;\n end do;\n end do;\n if prntf lg then\n print(``);\n print(convert(p,matrix));\n \+ print(``);\n print(convert(q,matrix));\n print(`` );\n end if;\n if plt and indets(p[0,0],name)=\{x\} and inde ts(q[0,0],name)=\{x\} then \n return [p[0,0],q[0,0],x=0..1];\n else\n return p[0,0],q[0,0];\n end if;\n else\n \+ for i from 1 to np do\n if not type(pts[i],complexcons) th en\n nmrc := false; # not numeric\n break;\n \+ end if;\n end do;\n if flt then pp := map(evalf,pts)\n \+ elif not nmrc then pp := map(normal,pts)\n else pp := pts en d if;\n\n if prntflg then\n p := array(0..n,0..n);\n \+ \n for i from 0 to n do p[i,n] := pp[i+1] end do;\n f or j from 0 to n-1 do\n for i from 0 to n do p[i,j] := 0 en d do;\n end do;\n else\n p := table(); \n \+ for i from 0 to n do p[i,n] := pp[i+1] end do; \n end if;\n \n # de Casteljau's algorithm\n for j from 1 to n do\n \+ for i from 0 to n-j do \n p[i,n-j] := (1-x)*p[i,np- j]+x*p[i+1,np-j];\n if not nmrc then p[i,n-j] := normal(p[i ,n-j]) end if;\n end do;\n end do;\n if prntflg then \n print(``);\n print(convert(p,matrix));\n pr int(``);\n end if;\n if plt and indets(p[0,0],name)=\{x\} th en\n return [x,p[0,0],x=0..1];\n else\n return p[ 0,0];\n end if;\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given i n the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "ca steljau" }{TEXT -1 36 ": examples of Bernstein polynomials " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "If the 1st argument is a list of " }{XPPEDIT 18 0 "n+1" " 6#,&%\"nG\"\"\"F%F%" }{TEXT -1 18 " numerical values " }{XPPEDIT 18 0 "[c[0],c[1],` . . . `,c[n]]" "6#7&&%\"cG6#\"\"!&F%6#\"\"\"%(~.~.~.~G&F %6#%\"nG" }{TEXT -1 40 ", and 2nd argument is a numerical value " } {TEXT 283 1 "x" }{TEXT -1 7 ", then " }{TEXT 0 9 "casteljau" }{TEXT -1 38 " computes the corresponding value of " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[0]*B[0,n](x)+c[1]*B[1,n](x)+` . . . ` +c[n]*B[n,n](x);" "6#,**&&%\"cG6#\"\"!\"\"\"-&%\"BG6$F(%\"nG6#%\"xGF)F )*&&F&6#F)F)-&F,6$F)F.6#F0F)F)%(~.~.~.~GF)*&&F&6#F.F)-&F,6$F.F.6#F0F)F )" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "The option \"" }{TEXT 284 9 "info=true" }{TEXT -1 58 " \" causes the intermediate computed values to be displayed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "pt s := [0,3,2,0];\ncasteljau(pts,0.7,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&\"\"!\"\"$\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7& $\"%\\9!\"$$\"$C#!\"#$\"#@!\"\"$\"\"!F27&F2$\"$6\"F-$\"#BF0$\"\"$F27&F 2F2$\"\"'F0$\"\"#F27&F2F2F2F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%\\9!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We can obtain a sequence \+ of points along the curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=c[0]*B[0,n](x)+c[1]*B[1,n](x)+` . . . `+c[n]*B[n,n](x )" "6#/%\"yG,**&&%\"cG6#\"\"!\"\"\"-&%\"BG6$F*%\"nG6#%\"xGF+F+*&&F(6#F +F+-&F.6$F+F06#F2F+F+%(~.~.~.~GF+*&&F(6#F0F+-&F.6$F0F06#F2F+F+" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 14 "and plot them." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pts := [0,3,2,0];\n[seq([i*0.05,casteljau(pts,i*0.05)],i=0..20)]; \nplot(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&\"\"!\"\"$\"\"# F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#777$$\"\"!F&F%7$$\"\"&!\"#$\"'v. U!\"'7$$\"#5F*$\"'+IyF-7$$\"#:F*$\"(D,4\"F-7$$\"#?F*$\"(+SM\"F-7$$\"#D F*$\"(voa\"F-7$$\"#IF*$\"(+5q\"F-7$$\"#NF*$\"(D'3=F-7$$\"#SF*$\"(+?(=F -7$$\"#XF*$\"(vL*=F-7$$\"#]F*$\"(+](=F-7$$\"#bF*$\"(D\">=F-7$$\"#gF*$ \"(+!G$\"3******* ****\\P?%!#=7$$\"3/+++++++5F0$\"3H++++++IyF07$$\"3%**************\\\"F 0$\"3+++++]7!4\"!#<7$$\"35+++++++?F0$\"33++++++W8F;7$$\"3++++++++DF0$ \"3+++++](oa\"F;7$$\"3))**************HF0$\"32++++++,=F;7$$\"3w**************fF0$\"3)************zs\"F;7$ $\"3A+++++++lF0$\"3%*********\\(Qg\"F;7$$\"3a**************pF0$\"31+++ +++\\9F;7$$\"3++++++++vF0$\"3+++++]il7F;7$$\"3U+++++++!)F0$\"31++++++c 5F;7$$\"3w*************\\)F0$\"3n*********\\PA)F07$$\"3A+++++++!*F0$\" 3[************pcF07$$\"3a*************\\*F0$\"3C+++++D@HF07$$\"\"\"F)F (-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q!6\"Fcr-%%VIEWG6$%(D EFAULTGFhr" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "If the 2nd argument \+ is an \"unknown\" parameter, we obtain the polynomial " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[0]*B[0,n](x)+c[1]*B[1,n](x)+` \+ . . . `+c[n]*B[n,n](x);" "6#,**&&%\"cG6#\"\"!\"\"\"-&%\"BG6$F(%\"nG6#% \"xGF)F)*&&F&6#F)F)-&F,6$F)F.6#F0F)F)%(~.~.~.~GF)*&&F&6#F.F)-&F,6$F.F. 6#F0F)F)" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 31 "converted t o the standard form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[0]+a[1]*x+` . . . `+a[n]*x^n" "6#,*&%\"aG6#\"\"!\"\"\"*&&F%6#F(F( %\"xGF(F(%(~.~.~.~GF(*&&F%6#%\"nGF()F,F1F(F(" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "pts := [0,3,2,0];\ncasteljau(pts,x, info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&\"\"!\"\"$\"\" #F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&,(*&\"\"*\"\"\"%\"xGF+F+*&\"#7F+)F,\"\" #F+!\"\"*&\"\"$F+)F,F3F+F+,&*&\"\"'F+F,F+F+*&\"\"%F+F/F+F1,$*&F3F+F,F+ F+\"\"!7&F<,(F3F+*&F0F+F,F+F1*$F/F+F1,&F3F+F,F1F37&F " 0 "" {MPLTEXT 1 0 67 "pts := [0,3,2,0]; \ncurve := casteljau(pts,t,plot=true);\nplot(curve);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&\"\"!\"\"$\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&curveG7%%\"tG,(*&\"\"*\"\"\"F&F*F**&\"#7F*)F&\"\"#F* !\"\"*&\"\"$F*)F&F1F*F*/F&;\"\"!F*" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$\"\"!F)F(7$$\"3ILLL3x&)*3\"!#> $\"3;u\"RE&Hdm'*F-7$$\"3emmm;arz@F-$\"3&>>$o[1/0>!#=7$$\"3.++D\"y%*z7$ F-$\"3yPc!*[7q)p#F57$$\"3[LL$e9ui2%F-$\"3'>vwFa'GrMF57$$\"3nmmm\"z_\"4 iF-$\"3rwNeBhxK^F57$$\"3[mmmT&phN)F-$\"3dP3[$)*\\,q'F57$$\"3CLLe*=)H\\ 5F5$\"3!3V]o$36d\")F57$$\"3gmm\"z/3uC\"F5$\"3(eTcwBqwT*F57$$\"3%)***\\ 7LRDX\"F5$\"3Uo$fn%\\Hj5!#<7$$\"3]mm\"zR'ok;F5$\"3![v%o-b^z6FY7$$\"3w* **\\i5`h(=F5$\"3qpRKOc&fG\"FY7$$\"3WLLL3En$4#F5$\"3!zxO.;FY7$$\"3)******\\nHi#HF5$\"3u+ \"G\"H'Q7o\"FY7$$\"3jmm\"z*ev:JF5$\"3AV(\\dzq*HY2aF 5$\"3#)p'4=T\">K=FY7$$\"39mm;zXu9cF5$\"3SS&o_zZ7!=FY7$$\"3l******\\y)) GeF5$\"3?7L&fv7Iw\"FY7$$\"3'*)***\\i_QQgF5$\"3o54q1 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 95 "In this example we construct the Bernstein polynomial of degree 8 \+ associated with the function " }{XPPEDIT 18 0 "f(x)=x/(1+x)" "6#/-%\"f G6#%\"xG*&F'\"\"\",&F)F)F'F)!\"\"" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "n := 8 ;\nf := x -> x/(1+x);\nxvals := [seq(k/n,k=0..n)];\nyvals := map(f,xva ls);\ncasteljau(yvals,x);\ng := unapply(%,x);\npts := zip((x,y)->[x,y] ,xvals,yvals):\nplot([g(x),f(x),pts],x=0..1,color=[red,magenta,blue], \n style=[line$2,point],symbol=circle,linestyle=[1,2],thickness=[ 2,1],\n legend=[`g(x)`,`f(x)`,`f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F.F.F-F.!\"\"F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7+\"\"!#\"\"\"\"\")#F(\"\"%# \"\"$F)#F(\"\"##\"\"&F)#F-F+#\"\"(F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7+\"\"!#\"\"\"\"\"*#F(\"\"&#\"\"$\"#6#F(F-#F+\"#8#F-\" \"(#F3\"#:#F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,29$#\"\")\"\"**&#\"#G\"#X\"\"\"*$)F-\"\"#F5F 5!\"\"*&#\"#c\"$l\"F5)F-\"\"$F5F5*&#\"#9\"#**F5*$)F-\"\"%F5F5F9*&#F<\" %(G\"F5)F-\"\"&F5F5*&#FF\"$H%F5*$)F-\"\"'F5F5F9*&#F/\"%NkF5)F-\"\"(F5F 5*&#F5\"&qG\"F5*$)F-F/F5F5F9F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 467 364 364 {PLOTDATA 2 "6(-%'CURVESG6(7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\" 3HegDd.J3>F-7$$\"3[LL$e9ui2%F-$\"3x=#3$*[EA_$F-7$$\"3nmmm\"z_\"4iF-$\" 3-#z6aMwsG&F-7$$\"3[mmmT&phN)F-$\"3*e1$*oumB,(F-7$$\"3CLLe*=)H\\5!#=$ \"3o/?z?>cz')F-7$$\"3gmm\"z/3uC\"FB$\"3CIYGXdC=5FB7$$\"3%)***\\7LRDX\" FB$\"3KAI/3Vmp6FB7$$\"3]mm\"zR'ok;FB$\"3A]V\")em\">K\"FB7$$\"3w***\\i5 `h(=FB$\"3qAvr@5Vp9FB7$$\"3WLLL3En$4#FB$\"3y%)f>s:*oh\"FB7$$\"3qmm;/RE &G#FB$\"3E'HAcE,Lu\"FB7$$\"3\")*****\\K]4]#FB$\"3g>r#[UW=)=FB7$$\"3$** ****\\PAvr#FB$\"3t0Z**Hr2FFB7$$\"3A++D\"=lj;%FB$\"3YfG)\\q#>JGF B7$$\"31++vV&RY2aFB$\"3#QGh9R%*3U $FB7$$\"39mm;zXu9cFB$\"3*=pa(oG36NFB7$$\"3l******\\y))GeFB$\"3f*Q2[2p> g$FB7$$\"3'*)***\\i_QQgFB$\"3Y(40(Hbq)o$FB7$$\"3@***\\7y%3TiFB$\"3el:L &eT1x$FB7$$\"35****\\P![hY'FB$\"3I\"4LF8-%fQFB7$$\"3kKLL$Qx$omFB$\"3ZX %>9CJs$RFB7$$\"3!)*****\\P+V)oFB$\"3t/\"H[!>Q=SFB7$$\"3?mm\"zpe*zqFB$ \"3E&zjn=J-4%FB7$$\"3%)*****\\#\\'QH(FB$\"3MG5M*H/q;%FB7$$\"3GKLe9S8& \\(FB$\"3$**eCbP(fPUFB7$$\"3R***\\i?=bq(FB$\"3/EjX0]t4VFB7$$\"3\"HLL$3 s?6zFB$\"3Qut*4ey'yVFB7$$\"3a***\\7`Wl7)FB$\"3G+!*\\X*G#\\WFB7$$\"3#pm mm'*RRL)FB$\"39xyi,7l:XFB7$$\"3Qmm;a<.Y&)FB$\"3y9_X1d2#e%FB7$$\"3=LLe9 tOc()FB$\"3[%RJM7&\\YYFB7$$\"3u******\\Qk\\*)FB$\"3$y--M]\\Wq%FB7$$\"3 CLL$3dg6<*FB$\"3m6?W=OXpZFB7$$\"3ImmmmxGp$*FB$\"39Br\"*[vME[FB7$$\"3A+ +D\"oK0e*FB$\"3]p`7&y\\d)[FB7$$\"3A++v=5s#y*FB$\"3FW4^*4B9%\\FB7$$\"\" \"F)$\"3K)*************\\FB-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&STYL EG6#%%LINEG-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#Fcz-%'LEGENDG6#%%g(x)G-F $6(7SF'7$F+$\"3r'RM#\\t@L@F-7$F1$\"31)=O***=i;RF-7$F6$\"3G'GZ7.ch%eF-7 $F;$\"3k(**)\\i;w6xF-7$F@$\"3E>'*)HO9l\\*F-7$FF$\"3j.2-xE146FB7$FK$\"3 7oBn@?Jo7FB7$FP$\"3Z/$FB7$F`r$\"3IB.`!3BT9$FB7$Fer$\"3#p-VWQ&QNKFB7 $Fjr$\"3yjbuz)Q6L$FB7$F_s$\"33j!o!y6xFMFB7$Fds$\"313w[Z$Q'4NFB7$Fis$\" 3M(e^2i'z&f$FB7$F^t$\"3Qg-rgnV#o$FB7$Fct$\"3+'e&)oNe\\w$FB7$Fht$\"3m7R N:exUQFB7$F]u$\"3Xc^iIU$p#RFB7$Fbu$\"3\"\\(Hz%z:1+%FB7$Fgu$\"3yWjQ]#Qt 2%FB7$F\\v$\"3y[GQ0\\=XTFB7$Fav$\"3dkRhvDgq3*>x%oYFB7$Fix$\"3;#QVa' f&Gs%FB7$F^y$\"3k)>%G-<$Qy%FB7$Fcy$\"3p,o^^v=P[FB7$Fhy$\"3'fZsEk')G*[F B7$F]z$\"3Y1t@YO3X\\FB7$Fbz$\"3++++++++]FB-Fgz6&FizFjzF(FjzF][l-Fb[lFg [l-Ff[lFc[l-Fi[l6#%%f(x)G-F$6(7+F'7$$\"3+++++++]7FB$\"3066666666FB7$$ \"3++++++++DFB$\"35+++++++?FB7$$\"3+++++++]PFB$\"32FFFFFFFFFB7$F]el$\" 3:LLLLLLLLFB7$$\"3+++++++]iFB$\"3OYQ:YQ:YQFB7$$\"3++++++++vFB$\"3[&G9d G9dG%FB7$$\"3+++++++]()FB$\"3ummmmmmmYFBF\\el-Fgz6&FizF(F(Fjz-F^[l6#%& POINTGFa[lFe[l-Fi[l6#%,f(x)~pointsG-%+AXESLABELSG6$Q\"x6\"Q!6\"-%'SYMB OLG6#%'CIRCLEG-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "g(x)" "f(x)" "f(x) points" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B(n,x) = Sum(f(k/n)*B[k,n](x),k = 0 \+ .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$SumG6$*&-%\"fG6#*&%\"kG\"\"\"F'!\"\"F 2-&F%6$F1F'6#F(F2/F1;\"\"!F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(f(k/n)*matrix([[n], [k]])*(1-x )^(n-k)*x^k,k = 0 .. n);" "6#/%!G-%$SumG6$**-%\"fG6#*&%\"kG\"\"\"%\"nG !\"\"F.-%'matrixG6#7$7#F/7#F-F.),&F.F.%\"xGF0,&F/F.F-F0F.)F9F-F./F-;\" \"!F/" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 26 "gives the same polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "alias(C=binomial):\nn := 8;\nf := x -> x/(1+x);\n Sum('f(k/n)'*C(n,k)*(1-x)^(n-k)*x^k,k=0..n);\ncollect(value(%),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F. F.F-F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$**-%\"fG 6#*&%\"kG\"\"\"%\"nG!\"\"F,-%\"CG6$\"\")F+F,),&F,F,%\"xGF.,&F2F,F+F.F, )F5F+F,/F+;\"\"!F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*&#\"\"\"\"&qG \"F&*$)%\"xG\"\")F&F&!\"\"*&#F+\"%NkF&*$)F*\"\"(F&F&F&*&#\"\"%\"$H%F&* $)F*\"\"'F&F&F,*&#\"#c\"%(G\"F&*$)F*\"\"&F&F&F&*&#\"#9\"#**F&*$)F*F5F& F&F,*&#F<\"$l\"F&*$)F*\"\"$F&F&F&*&#\"#G\"#XF&*$)F*\"\"#F&F&F,*&#F+\" \"*F&F*F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 9 "bernstein" }{TEXT -1 18 " can also be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := x -> x/(1+x):\nbernstein(8,f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*&#\"\"\"\"&qG\"F&*$)%\"xG\"\")F&F&!\"\"*& #F+\"%NkF&*$)F*\"\"(F&F&F&*&#\"\"%\"$H%F&*$)F*\"\"'F&F&F,*&#\"#c\"%(G \"F&*$)F*\"\"&F&F&F&*&#\"#9\"#**F&*$)F*F5F&F&F,*&#F<\"$l\"F&*$)F*\"\"$ F&F&F&*&#\"#G\"#XF&*$)F*\"\"#F&F&F,*&#F+\"\"*F&F*F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "n := 8 ;\nf := x -> x/(1+x);\nxvals := [seq(k/n,k=0..n)];\nyvals := map(f,xva ls);\ncasteljau(yvals,x,plot=true);\nplot(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 95 "In this example we construct the Bernstein polynomial of degree 8 \+ associated with the function " }{XPPEDIT 18 0 "f(x) = 1/2-abs(x-1/2); " "6#/-%\"fG6#%\"xG,&*&\"\"\"F*\"\"#!\"\"F*-%$absG6#,&F'F**&F*F*F+F,F, F," }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"! \"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "n := 8;\nf := x -> 1/2-abs(x-1/2); \nxvals := [seq(k/n,k=0..n)];\nyvals := map(f,xvals);\ncasteljau(yvals ,x);\ng := unapply(%,x);\npts := zip((x,y)->[x,y],xvals,yvals):\nplot( [g(x),f(x),pts],x=0..1,color=[red,magenta,blue],\n style=[line$2, point],symbol=circle,linestyle=[1,2],thickness=[2,1],\n legend=[ `g(x)`,`f(x)`,`f(x) points`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" nG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,&#\"\"\"\"\"#F.-%$absG6#,&9$F.#F.F/!\"\"F6F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7+\"\"!#\"\"\"\"\")#F(\"\"%# \"\"$F)#F(\"\"##\"\"&F)#F-F+#\"\"(F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7+\"\"!#\"\"\"\"\")#F(\"\"%#\"\"$F)#F(\"\"#F,F*F'F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,%\"xG\"\"\"*&\"#9F%)F$\"\"&F%!\"\"*& \"#GF%)F$\"\"'F%F%*&\"#?F%)F$\"\"(F%F**&F)F%)F$\"\")F%F%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,,9$\" \"\"*&\"#9F.)F-\"\"&F.!\"\"*&\"#GF.)F-\"\"'F.F.*&\"#?F.)F-\"\"(F.F3*&F 2F.)F-\"\")F.F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 467 364 364 {PLOTDATA 2 "6(-%'CURVESG6(7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3SZ.yB)3 (z@F-7$$\"3[LL$e9ui2%F-$\"3m^g8h!Hh2%F-7$$\"3nmmm\"z_\"4iF-$\"3hdjGfT, 3iF-7$$\"3[mmmT&phN)F-$\"3wTs>tKO^$)F-7$$\"3CLLe*=)H\\5!#=$\"3uq(*[2Q' y/\"FB7$$\"3gmm\"z/3uC\"FB$\"3TB2AMN9W7FB7$$\"3%)***\\7LRDX\"FB$\"3!p0 Ads`eW\"FB7$$\"3]mm\"zR'ok;FB$\"3W*>cO\"RorR$FB7$$\"3A++D\"=lj;%FB$\"3#449jP#)G[$FB7$$\"31++vV &RY2aFB$\"37q#H$eZh'f$FB7$$\" 39mm;zXu9cFB$\"3udf8Ybw]NFB7$$\"3l******\\y))GeFB$\"3C$G#G3=c%[$FB7$$ \"3'*)***\\i_QQgFB$\"3K`WZzf%>S$FB7$$\"3@***\\7y%3TiFB$\"3j*HEpV+gI$FB 7$$\"35****\\P![hY'FB$\"3mcUD'y?@=$FB7$$\"3kKLL$Qx$omFB$\"31z205DMcIFB 7$$\"3!)*****\\P+V)oFB$\"3T\\xj$y@#3HFB7$$\"3?mm\"zpe*zqFB$\"3'*G^wu3% Gw#FB7$$\"3%)*****\\#\\'QH(FB$\"3w>A,b]5$f#FB7$$\"3GKLe9S8&\\(FB$\"3W6 X]vOPCCFB7$$\"3R***\\i?=bq(FB$\"3`T#[xp$**RAFB7$$\"3\"HLL$3s?6zFB$\"3+ ]uEy?4`?FB7$$\"3a***\\7`Wl7)FB$\"3eRu#p<2<&=FB7$$\"3#pmmm'*RRL)FB$\"3? v)*y))RR`;FB7$$\"3Qmm;a<.Y&)FB$\"3[sTn!y^sW\"FB7$$\"3=LLe9tOc()FB$\"3% >qOOm9/C\"FB7$$\"3u******\\Qk\\*)FB$\"3oM_MY[\"*[5FB7$$\"3CLL$3dg6<*FB $\"3#ycwBBtPG)F-7$$\"3ImmmmxGp$*FB$\"3,DK!>\\$*eI'F-7$$\"3A++D\"oK0e*F B$\"3g8)4w<1X>%F-7$$\"3A++v=5s#y*FB$\"3OgyxALys@F-7$$\"\"\"F)F(-%'COLO URG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"#-%* LINESTYLEG6#Fcz-%'LEGENDG6#%%g(x)G-F$6(7UF'7$F+F+7$F1$\"3EOL$e9ui2%F-7 $F6F67$F;$\"3EpmmT&phN)F-7$F@$\"35LLe*=)H\\5FB7$FFFF7$FK$\"37++DJ$RDX \"FB7$FPFP7$FUFU7$FZFZ7$FinFin7$F^oF^o7$FcoFco7$FhoFho7$F]pF]p7$FbpFbp 7$FgpFgp7$F\\qF\\q7$FaqFaq7$FfqFfq7$F[rF[r7$F`rF`r7$FerFer7$$\"3%***\\ (=7O*))[FBF]^l7$FjrFjr7$$\"3@m;/^7I0^FB$\"3yL$e*[()p%*[FB7$F_s$\"3;ML3 -;Y%y%FB7$Fds$\"3]++D\"3QDf%FB7$Fis$\"3'QLL3Ub_Q%FB7$F^t$\"3O+++]@6rTF B7$Fct$\"3/,+]PZhhRFB7$Fht$\"3y++v=_\"*ePFB7$F]u$\"3*3++D'>&Q`$FB7$Fbu $\"3Pnmm;EiJLFB7$Fgu$\"3?+++D'*p:JFB7$F\\v$\"3zLL3-8/?HFB7$Fav$\"3<+++ v]81FFB7$Ffv$\"3snmT&)f'[]#FB7$F[w$\"3g++v$z\"[%H#FB7$F`w$\"33nmm\"z#z )3#FB7$Few$\"3Y++voaXt=FB7$Fjw$\"33LLLL+1m;FB7$F_x$\"3iLL$eCoRX\"FB7$F dx$\"3$om;aoKOC\"FB7$Fix$\"3E+++]hN]5FB7$F^y$\"3anmm\"H%R)G)F-7$Fcy$\" 30PLLLB72jF-7$Fhy$\"3i(***\\(=tY>%F-7$F]z$\"3#y***\\7)*ys@F-Faz-Fez6&F gzFhzF(FhzF[[l-F`[lFe[l-Fd[lFa[l-Fg[l6#%%f(x)G-F$6(7+F'7$$\"3+++++++]7 FBFecl7$$\"3++++++++DFBFhcl7$$\"3+++++++]PFBF[dl7$$\"3++++++++]FBF^dl7 $$\"3+++++++]iFBF[dl7$$\"3++++++++vFBFhcl7$$\"3+++++++]()FBFeclFaz-Fez 6&FgzF(F(Fhz-F\\[l6#%&POINTGF_[lFc[l-Fg[l6#%,f(x)~pointsG-%+AXESLABELS G6$Q\"x6\"Q!6\"-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "g(x)" "f(x)" "f(x) \+ points" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 9 "bernstein" }{TEXT -1 18 " can al so be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "f := x -> 1/2-abs(x-1/2):\nbernstein(8,f,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,%\"xG\"\"\"*&\"#9F%)F$\"\"&F%!\"\"*& \"#GF%)F$\"\"'F%F%*&\"#?F%)F$\"\"(F%F**&F)F%)F$\"\")F%F%" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "casteljau" }{TEXT -1 48 ": examples where the input is a list of points " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "If the 1st arument is a l ist of points and the 2nd argument is a numerical value, then " } {TEXT 0 9 "casteljau" }{TEXT -1 64 " gives the two coordinates of the \+ associated point on the curve." }}{PARA 0 "" 0 "" {TEXT -1 12 "The opt ion \"" }{TEXT 284 9 "info=true" }{TEXT -1 58 "\" causes the intermedi ate computed values to be displayed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "pts := [[0,0],[1,2],[3,3], [6,0]];\ncasteljau(pts,0.7,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&7$\"\"!F'7$\"\"\"\"\"#7$\"\"$F,7$\"\"'F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrix G6#7&7&$\"%qN!\"$$\"$*=!\"#$\"\"(!\"\"$\"\"!F27&F2$\"$H%F-$\"#CF0$\"\" \"F27&F2F2$\"#^F0$\"\"$F27&F2F2F2$\"\"'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7& $\"%, " 0 "" {MPLTEXT 1 0 82 "pts := [[0,0 ],[1,2],[3,3],[6,0]];\n[seq([casteljau(pts,i*0.05)],i=0..20)];\nplot(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&7$\"\"!F'7$\"\"\"\"\"#7 $\"\"$F,7$\"\"'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#777$$\"\"!F&F%7$$ \"'+v:!\"'$\"'D@HF*7$$\"'++LF*$\"'+qcF*7$$\"'+v^F*$\"'vB#)F*7$$\"'++sF *$\"(+g0\"F*7$$\"'+v$*F*$\"(DcE\"F*7$$\"(++<\"F*$\"(+!\\9F*7$$\"(+vT\" F*$\"(vQg\"F*7$$\"(++o\"F*$\"(+!GF*$\"(D\">=F*7$$\"(++D#F *$\"(+](=F*7$$\"(+vb#F*$\"(vL*=F*7$$\"(++)GF*$\"(+?(=F*7$$\"(+v@$F*$\" (D'3=F*7$$\"(++d$F*$\"(+5q\"F*7$$\"(+v$RF*$\"(voa\"F*7$$\"(++K%F*$\"(+ SM\"F*7$$\"(+vr%F*$\"(D,4\"F*7$$\"(++8&F*$\"'+IyF*7$$\"(+vb&F*$\"'v.UF *7$$\"(+++'F*F%" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$777$$\"\"!F)F(7$$\"3-++++++v:!#=$\"3C+++++D@HF-7$$\"3;+ ++++++LF-$\"3[************pcF-7$$\"3g***********\\<&F-$\"3n*********\\ PA)F-7$$\"3u*************>(F-$\"31++++++c5!#<7$$\"3+++++++v$*F-$\"3+++ ++]il7F?7$$\"3#*************p6F?$\"31++++++\\9F?7$$\"3)***********\\<9 F?$\"3%*********\\(Qg\"F?7$$\"3%*************z;F?$\"3)************zs\" F?7$$\"3-+++++]d>F?$\"35++++]7>=F?7$$\"3+++++++]AF?$\"3+++++++v=F?7$$ \"35+++++]dDF?$\"3-++++]P$*=F?7$$\"3))************zGF?$\"35++++++s=F?7 $$\"3!)**********\\ " 0 "" {MPLTEXT 1 0 61 "pts := [[0,0],[1,2 ],[3,3],[6,0]];\ncasteljau(pts,t,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&7$\"\"!F'7$\"\"\"\"\"#7$\"\"$F,7$\"\"'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'matrixG6#7&7&,&%\"tG\"\"$*&F*\"\"\")F)\"\"#F,F,,&F)F.*$F-F,F,F)\" \"!7&F1,(F,F,*&\"\"%F,F)F,F,F0F,,&F,F,*&F.F,F)F,F,F,7&F1F1,&F*F,*&F*F, F)F,F,F*7&F1F1F1\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&,(%\"tG\"\"'*&\"\"$\"\"\")F) \"\"#F-!\"\"*&F,F-)F)F,F-F0,&F)\"\"%*$F.F-F0,$F)F/\"\"!7&F7,(F/F-*&F/F -F)F-F-*&F4F-F.F-F0,&F/F-F)F-F/7&F7F7,&F,F-*&F,F-F)F-F0F,7&F7F7F7F7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$,&%\"tG\"\"$*&F%\"\"\")F$\"\"#F'F',(F$\"\"'*&F%F'F(F'!\"\"*&F%F')F$F %F'F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The option \"" }{TEXT 284 9 "plot=true" }{TEXT -1 23 "\" causes the p rocedure " }{TEXT 0 9 "casteljau" }{TEXT -1 52 " to construct suitable input information needed for " }{TEXT 0 4 "plot" }{TEXT -1 26 " to pl ot the Bezier curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "pts := [[0,0],[1,2],[3,3],[6,0]];\ncurve := casteljau(pts,t,plot=true);\nplot([curve,pts,pts],style=[line$2,po int],color=[red,brown,black],\n symbol=circle,linestyle=[1,2],thick ness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&7$\"\"!F'7$\"\"\" \"\"#7$\"\"$F,7$\"\"'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&curveG7% ,&*&\"\"$\"\"\"%\"tGF)F)*&F(F))F*\"\"#F)F),(*&\"\"'F)F*F)F)*&F(F)F,F)! \"\"*&F(F))F*F(F)F2/F*;\"\"!F)" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6&7U7$$\"\"!F)F(7$$\"3'*H'H*G5o\"o'!#>$\" 3#H*>&o3lKH\"!#=7$$\"3)z'f[w-ts7F0$\"3O4MH7X)QR#F07$$\"3%eyqEd1%y>F0$ \"3\"\\\\i!)Q\\Eg$F07$$\"3uKssqvK;FF0$\"3]'*H)yy?ny%F07$$\"3ud)[fd-#yM F0$\"3uw_!4(>#3$fF07$$\"3I!=_^>K!4UF0$\"3_!o#3C-TfpF07$$\"3Me%)RY\"z0* \\F0$\"3q`\\%oQV\"FY$\"3S14mxMx7;FY7$$\"3!>m!3lY.\\:FY$\"3/%[i2aU4n \"FY7$$\"31\"[2J@XMl\"FY$\"3P&Hl]y)=<vmqRqC#FY$\"3s%=V4)ziu=FY 7$$\"37%HzFu;2Q#FY$\"3K\\r4!3^w)=FY7$$\"3)H%pq2zX*\\#FY$\"3q:c!f![!H*= FY7$$\"3rV%GW2%=IEFY$\"3#*R#*3qb1#*=FY7$$\"3WGj.iV%zw#FY$\"3eW]'Q)*GR) =FY7$$\"31:85w%y`!HFY$\"3#4^#H+$FY$\"3uFV%f*pG9=FY7$$\"3&\\a'zA4`MLFY$\"3y)yR5PNux\"F Y7$$\"3Kan4iy4([$FY$\"3Li0.8t&*H;;FY7$$\"3W@>F@7&Q$RFY$\"3kjM1*>1'[:FY7$$\"3 k00o'y0H4%FY$\"3uPn<=\\^p9FY7$$\"3yebHZ\"y4D%FY$\"3-!zk)Guo$Q\"FY7$$\" 3%znE*R^=>WFY$\"31fZgNil%G\"FY7$$\"3U?,/^'=Qe%FY$\"39mWTMqB!=\"FY7$$\" 3E\\na)G\\[v%FY$\"3ep-)=v4T1\"FY7$$\"3u+@\"3DHr#\\FY$\"3+m9'4O]WR*F07$ $\"3+%Gbi!px(3&FY$\"3wq.w+=2k\")F07$$\"39S-tdPlu_FY$\"3?]lIdoE_mF07$$ \"3l*GQw#HHWaFY$\"3ID(G\\Zzl?&F07$$\"3Xz*p!)*zvFcFY$\"37)[C[mwic$F07$$ \"3uF#z)3]];dFY$\"3;s`(yr#pWFF07$$\"3K\"zX:Ale!eFY$\"37iVlxj;**=F07$$ \"3b(>0Y`yD!fFY$\"3Bhfrb$3jj*F-7$$\"\"'F)F(-%'COLOURG6&%$RGBG$\"*++++ \"!\")F(F(-%&STYLEG6#%%LINEG-%*LINESTYLEG6#\"\"\"-F$6&7&F'7$$F]\\lF)$ \"\"#F)7$$\"\"$F)Ff\\lF\\[l-F`[l6&Fb[l$\")#)eqkFe[l$\"))eqk\"Fe[lF\\]l Ff[l-F[\\l6#Fd\\l-F$6&F`\\l-F`[l6&Fb[lF)F)F)-Fg[l6#%&POINTGFj[l-%*THIC KNESSGF_]l-%+AXESLABELSG6$Q!6\"F\\^l-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$%(D EFAULTGFe^l" 1 2 4 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 93 "In this example we plot the Bezier curve associate d with the 6 vertices of a regular hexagon." }}{PARA 0 "" 0 "" {TEXT -1 45 "There are actually 7 points because the point" }{XPPEDIT 18 0 " ``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 64 " appears as both the fir st and last point in the list of points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "h := Pi/3; \npts := \+ [seq([cos(i*h),sin(i*h)],i=0..6)];\ncasteljau(pts,x,plot=true);\nplot( [%,pts,pts],style=[line$2,point],thickness=2,\n symbol=circle,color =[red,brown,black],linestyle=[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"hG,$%#PiG#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG 7)7$\"\"\"\"\"!7$#F'\"\"#,$*$-%%sqrtG6#\"\"$F'F*7$#!\"\"F+F,7$F4F(7$F3 ,$F-F37$F*F7F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,0\"\"\"F%*&\"\"$F %%\"xGF%!\"\"*&\"#?F%)F(F'F%F%*&#\"#:\"\"#F%*$)F(\"\"%F%F%F)*&#F/F0F%* $)F(F0F%F%F)*&F'F%)F(\"\"&F%F)*$)F(\"\"'F%F%,**&F(F%-%%sqrtG6#F'F%F'*& #F/F0F%*&F7F%F@F%F%F)*(#F/F0F%F2F%F@F%F%*(F'F%F9F%F@F%F)/F(;\"\"!F%" } }{PARA 13 "" 1 "" {GLPLOT2D 378 340 340 {PLOTDATA 2 "6)-%'CURVESG6&7W7 $$\"\"\"\"\"!$F*F*7$$\"3S*\\ma5#Rk'*!#=$\"3=&oHZf'y3b!#>7$$\"3&Q?E'Qe] 7$*F/$\"3%f^fF0B42\"F/7$$\"31j!Q8KnU**)F/$\"3'RkF\">TP)\\\"F/7$$\"3E;7 GeN$em)F/$\"3_AqOq')f->F/7$$\"352j9Qu$[*yF/$\"3#HMiQmFus#F/7$$\"3K)=P( )pwB3(F/$\"3f9(G9@a5W$F/7$$\"3Um#)Hoi$zC'F/$\"3DeHzdW7PSF/7$$\"3!>z^;E G*faF/$\"3/Ma=\"Rn-\\%F/7$$\"3K@p]d0oPYF/$\"3Irak\"4\"Gh[F/7$$\"39YArI k'*)y$F/$\"3fHq!*)*4@V^F/7$$\"3=*HZd23H&HF/$\"3J[#\\/T+^K&F/7$$\"3$>'* [ADa:6#F/$\"3#fE)pPsX8aF/7$$\"39'R)HIU[#R\"F/$\"3>bI0hHN7aF/7$$\"3&QO' **HhHVhF2$\"3e,2A(yDvK&F/7$$!3y6=$zJY)p7F2$\"3oLut)=1)e^F/7$$!3+iW#=#> ^uzF2$\"3RiDIV\")pA\\F/7$$!35D^#Q+\"[k8F/$\"3_mN9k-l]YF/7$$!3yfOsou$=) >F/$\"3q^%*GhJ9iUF/7$$!3I=b>Tdn_CF/$\"3'*\\sWWBB$)QF/7$$!3cch![F?h$HF/ $\"3C()>80:'=R$F/7$$!3YqAQX>O.LF/$\"3;6wqw'Rv\"HF/7$$!3I1@&eI=yj$F/$\" 3'>Y7DSH:O#F/7$$!3#zX*yCXY()QF/$\"3oEmuS#RT!=F/7$$!3mG/PVK[uSF/$\"3IGO '>$zw+7F/7$$!3w740,M(*yTF/$\"36TsX4Y&[L'F27$$!3DC$oOWH(=UF/$\"3^%\\>=r zEW\"!#?7$$!3!y&*4LNw&zTF/$!397;'*z\"GoG'F27$$!3%Grq!yi(*ySF/$!3#Rv#=P A;#=\"F/7$$!34u8pFQ\\,RF/$!31GLOkRmmIUzQF/7$$!3a,z'f)H8d>F/$!37X9(GO\\)zUF/7$$! 31>RDL*=VO\"F/$!3m`U'3sQ2l%F/7$$!3/Wo`r2V#y(F2$!3)*>6h2CoI\\F/7$$!3Zk) H%[t66*)Fjs$!3eu'p2Yl'p^F/7$$\"3V*)=Z+V]0gF2$!3d*3,lf+_K&F/7$$\"3)>3K& yO]e8F/$!3Z`eRtM]5aF/7$$\"3.paUrO;I@F/$!3oZ#\\PTPDT&F/7$$\"3oG;(RqxM'H F/$!36y-AQeQB`F/7$$\"3'ejk!4[\\$y$F/$!3@^y/VuqW^F/7$$\"3-kuV_!\\>j%F/$ !3!)*e$)G g_iFF/7$$\"3?)*=mv'eTi)F/$!3%eSy!=@W^>F/7$$\"3QoPmzbBv*)F/$!3\"yq%oxZx A:F/7$$\"3KRo;M,z9$*F/$!3i$H;\\h:x1\"F/7$$\"3%e%G->][l'*F/$!3u&oJI1s< \\&F2F'-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%&STYLEG6#%%LINEG-%*LINEST YLEG6#F)-F$6&7)F'7$$\"3++++++++]F/$\"3'fQWy.a-m)F/7$$!3++++++++]F/F\\] l7$$!\"\"F*F+7$F_]l$!3'fQWy.a-m)F/7$Fj\\lFe]lF'-Fi[l6&F[\\l$\")#)eqkF^ \\l$\"))eqk\"F^\\lF\\^lF_\\l-Fd\\l6#\"\"#-F$6&Fh\\l-Fi[l6&F[\\lF*F*F*- F`\\l6#%&POINTGFc\\l-%+AXESLABELSG6$Q!6\"F[_l-%*THICKNESSGF_^l-%'SYMBO LG6#%'CIRCLEG-%%VIEWG6$%(DEFAULTGFf_l" 1 2 4 1 10 2 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 51 "(a) Use de Casteljau' s algorithm via the procedure " }{TEXT 0 9 "casteljau" }{TEXT -1 70 " \+ to find the Bernstein polynomial B(5,x) associated with the function \+ " }{XPPEDIT 18 0 "f(x)=x/(1+x^2)" "6#/-%\"fG6#%\"xG*&F'\"\"\",&F)F)*$F '\"\"#F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(b) Plot the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 32 " and B(5,x) in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 35 "_____ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 51 "(a) Use de Ca steljau's algorithm via the procedure " }{TEXT 0 9 "casteljau" }{TEXT -1 34 " to find the Bernstein polynomial " }{XPPEDIT 18 0 "B(5,x)" "6# -%\"BG6$\"\"&%\"xG" }{TEXT -1 30 " associated with the function " } {XPPEDIT 18 0 "f(x) = x-x^5;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\"&!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(b) Plot the graph \+ of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 32 " and B(5,x) in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 35 "________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }