{VERSION 3 0 "APPLE_PPC_MAC" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times New Roman" 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Out put" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 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 } } {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Bicubic Surfaces" }} {PARA 0 "" 0 "" {TEXT -1 91 "This worksheet illustrates parametric bic ubic surfaces as developed in ''Computer Graphics\"" }}{PARA 0 "" 0 " " {TEXT -1 67 "by Foley and van Dam. It includes an example of a Herm ite Surface." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 " \+ J. Ross Beveridge 4/22/99" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(linalg):\nwit h(plots):\nreadlib(unassign):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Hermite Curves" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 "The Basis M atrix and Parameter Vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "M := matrix(4,4, [2,-2,1,1, -3,3,- 2,-1, 0,0,1,0, 1,0,0,0]);\nS := matrix(1,4, [s^3, s^2, s, 1]):\nT := \+ matrix(1,4, [t^3, t^2, t, 1]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" MG-%'matrixG6#7&7&\"\"#!\"#\"\"\"F,7&!\"$\"\"$F+!\"\"7&\"\"!F2F,F27&F, F2F2F2" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 57 "Define and Show the f our curves that control the surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "GG1 := matrix(4,3, [0,0,0, 0,1,1, 1,1,1, 0,1,0]):\nG G2 := matrix(4,3, [0,0,0, 1,0,1, 1,1,1, 1,0,1]):\nGG3 := matrix(4,3, [ 0,0,0, 1,1,1, 1,1,1, 1,1,1]):\nGG4 := matrix(4,3, [0,0,0, -1,-1,-1, -1 ,-1,-1, -1,-1,-1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 490 "Gt1 := eval m( T &* M &* GG1):\nGt2 := evalm( T &* M &* GG2):\nGt3 := evalm( T &* \+ M &* GG3):\nGt4 := evalm( T &* M &* GG4):\n[P1(t) = T * M * G1,\n P4(t ) = T * M * G2,\n R1(t) = T * M * G3,\n R4(t) = T * M * G4];\n[G1 = ev al(GG1), G2 = eval(GG2), G3 = eval(GG3), G4 = eval(GG4)];\n[x(s,t) = S * M * matrix(4,1, [P1[x](t), P2[x](t), R1[x](t), R4[x](t)]),\n y(s,t) = S * M * matrix(4,1, [P1[y](t), P2[y](t), R1[y](t), R4[y](t)]),\n z( s,t) = S * M * matrix(4,1, [P1[z](t), P2[z](t), R1[z](t), R4[z](t)])]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/-%#P1G6#%\"tG*(%\"TG\"\"\"%\"MG F+%#G1GF+/-%#P4GF'*(F*\"\"\"F,F2%#G2GF+/-%#R1GF'*(F*F2F,F2%#G3GF+/-%#R 4GF'*(F*F2F,F2%#G4GF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/%#G1G-%'ma trixG6#7&7%\"\"!F+F+7%F+\"\"\"F-7%F-F-F-7%F+F-F+/%#G2G-F'6#7&F*7%F-F+F -F.F5/%#G3G-F'6#7&F*F.F.F./%#G4G-F'6#7&F*7%!\"\"FAFAF@F@" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%/-%\"xG6$%\"sG%\"tG*(%\"SG\"\"\"%\"MGF,-%'matr ixG6#7&7#-&%#P1G6#F&6#F)7#-&%#P2GF6F77#-&%#R1GF6F77#-&%#R4GF6F7F,/-%\" yGF'*(F+\"\"\"F-FH-F/6#7&7#-&F56#FFF77#-&F;FOF77#-&F?FOF77#-&FCFOF7F,/ -%\"zGF'*(F+FHF-FH-F/6#7&7#-&F56#FenF77#-&F;F]oF77#-&F?F]oF77#-&FCF]oF 7F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "x1 := unapply(Gt1[1 ,1],t): y1 := unapply(Gt1[1,2],t): z1 := unapply(Gt1[1,3],t):\nx2 := u napply(Gt2[1,1],t): y2 := unapply(Gt2[1,2],t): z2 := unapply(Gt2[1,3], t):\nx3 := unapply(Gt3[1,1],t): y3 := unapply(Gt3[1,2],t): z3 := unapp ly(Gt3[1,3],t):\nx4 := unapply(Gt4[1,1],t): y4 := unapply(Gt4[1,2],t): z4 := unapply(Gt4[1,3],t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "sam ple := 100:\nPts1 := [seq([x1(i/sample), y1(i/sample), z1(i/sample)],i =0..sample)]:\nPts2 := [seq([x2(i/sample), y2(i/sample), z2(i/sample)] ,i=0..sample)]:\nPts3 := [seq([x3(i/sample), y3(i/sample), z3(i/sample )],i=0..sample)]:\nPts4 := [seq([x4(i/sample), y4(i/sample), z4(i/samp le)],i=0..sample)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "PLO T3D(CURVES(eval(Pts1),COLOR(RGB,1.0,0.0,0.0)),\n CURVES(eval(Pts 2),COLOR(RGB,0.0,0.0,1.0)),\n CURVES(eval(Pts3),COLOR(RGB,1.0,1. 0,0.0)),\n CURVES(eval(Pts4),COLOR(RGB,0.0,1.0,1.0)),\n TI TLE(CURVES),AXES(BOXED),LABELS([x,y,z]),FONT(HELVETICA,BOLD,14),THICKN ESS(4));" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6+-%'CURVESG6$7aq7%\"\"!F(F(7%#\"%,)*\"(+++\"#\" \"\"\"$+\"#\"&*45F,7%#\"%,C\"'+]7#F.\"#]#\"%\\DF57%#\"&F#GF,#\"\"$F/# \"&t3$F,7%#\"$w&\"&Dc\"#F.\"#D#\"$\\'FD7%#\"$h$\"%+!)#F.\"#?#\"$>%FL7% #\"%FmF5#F>F7#\"%BzF57%#\"&V0'F,#\"\"(F/#\"&dX(F,7%#\"%e5FD#\"\"#FF#\" %U8FD7%#\"&HX(F,#\"\"*F/#\"&rt*F,7%#\"#\")\"%+5#F.\"#5#\"$4\"Fio7%#\"& Jr)F,#\"#6F/#\"'p27F,7%#\"%_9FD#F>FF#\"%t?FD7%#\"&(R)*F,#\"#8F/#\"'.Z9 F,7%#\"&VH\"F5#FenF7#\"&2'>F57%#\"$n)FL#F>FN#\"%`8FL7%#\"%kF,7%#\"&H^\"F5#FcoF7#\"&@e#F57%#\"'f Y7F,#\"#>F/#\"'T#>#F,7%#\"#;\"$D\"#F.\"\"&#\"#HF\\t7%#\"'h58F,#\"#@F/# 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Gt1[1,3], \n Gt2[1,1], Gt2[1,2], Gt2[1,3], \n Gt3[1,1], Gt3[1,2], Gt3[1,3], \n \+ Gt4[1,1], Gt4[1,2], Gt4[1,3]]): \nQ(s,t) = S * M * G(t) ;\nQ(s,t) = \+ eval(S) * eval(M) * eval(GMH);\nQSH := evalm(S &* M &* GMH):\nx(s,t) = collect(QSH[1,1],\{t,s\});\ny(s,t) = collect(QSH[1,2],\{t,s\});\nz(s, t) = collect(QSH[1,3],\{t,s\});\nxh := unapply(collect(QSH[1,1],\{t,s \}), t,s):\nyh := unapply(collect(QSH[1,2],\{t,s\}), t,s):\nzh := unap ply(collect(QSH[1,3],\{t,s\}), t,s):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"QG6$%\"sG%\"tG*(%\"SG\"\"\"%\"MGF+-%\"GG6#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"QG6$%\"sG%\"tG*(-%'matrixG6#7#7&*$)F'\"\"$ \"\"\"*$)F'\"\"#F2F'\"\"\"F6-F+6#7&7&F5!\"#F6F67&!\"$F1F;!\"\"7&\"\"!F @F6F@7&F6F@F@F@F6-F+6#7&7%,(*$)F(F1F2F6*$)F(F5F2F;F(F6F(,(FGF>FIF6F(F6 7%F(FFF(7%F(F(F(7%,$F(F>FOFOF6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"xG6$%\"sG%\"tG,(*&,(*$)F'\"\"$\"\"\"\"\"#*$)F'F0F/!\"$\"\"\"F4F4)F(F .F/F4*&,(F,!\"%F1\"\"'!\"#F4F4)F(F0F/F4*&,(F1!\"\"F4F4F'F4F4F(F4F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6$%\"sG%\"tG,(*&,&*$)F'\"\"$\" \"\"!\"#*$)F'\"\"#F/F.\"\"\")F(F.F/F4*&,&F,\"\"%F1!\"'F4)F(F3F/F4*&,(F 1!\"\"F4F4F'F4F4F(F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"zG6$%\" sG%\"tG,(*&,(*$)F'\"\"$\"\"\"!\"#*$)F'\"\"#F/F.!\"\"\"\"\"F5)F(F.F/F5* &,(F,F3F1!\"$F5F5F5)F(F3F/F5*&,(F1F4F5F5F'F5F5F(F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "plot3d([xh(s,t),yh(s,t),zh(s,t)],s=0.0.. 1.0,t=0.0..1.0,axes=BOXED,labels=[x,y,z],labelfont=[HELVETICA,OBLIQUE, 14],\norientation=[-142,88],scaling=CONSTRAINED, color=[s,t,0]);" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%%MESHG6$W*^`r \":\":\"$0000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 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