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{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 68 "by Foley and van Dam.  It developes an example of a  Be
zier surface." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 105 "                                                        \+
              J. Ross Beveridge           4/27/99" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(linalg)
:\nwith(plots):\nreadlib(unassign):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Wa
rning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning
, new definition for trace" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 "D
eriving a Bezier Surface" }}{PARA 0 "" 0 "" {TEXT -1 101 "The derivati
on is motivated by extending cubic curves.  The essential idea is that
 of sweeping one 3D" }}{PARA 0 "" 0 "" {TEXT -1 49 "curve over a path \+
defined by four other curves.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 237 "Q(s,t) = S * Mb * Gb(t);\nS := matrix(1,4, [s^3, s^2, s, 1]):
\nT := matrix(1,4, [t^3, t^2, t, 1]):\nMb := matrix(4,4, [-1,3,-3,1,  \+
3,-6,3,0,  -3,3,0,0,  1,0,0,0]):\nQ(s,t) = eval(S) * eval(Mb) * matrix
(4,1, [Gb1(t), Gb2(t), Gb3(t), Gb4(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"QG6$%\"sG%\"tG*(%\"SG\"\"\"%#MbGF+-%#GbG6#F(F+" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%\"QG6$%\"sG%\"tG*(-%'matrixG6#7#7&*$)F'\"\"$
\"\"\"*$)F'\"\"#F2F'\"\"\"F6-F+6#7&7&!\"\"F1!\"$F67&F1!\"'F1\"\"!7&F<F
1F?F?7&F6F?F?F?F6-F+6#7&7#-%$Gb1G6#F(7#-%$Gb2GFH7#-%$Gb3GFH7#-%$Gb4GFH
F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "Gbi(t) = T * Mb * Gi;\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$GbiG6#%\"tG*(%\"TG\"\"\"%#MbGF*%#GiGF*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "GG1 := matrix(4,3, [0,0,1, 1,0,2, \+
2,0,3, 3,0,1]):\nGG2 := matrix(4,3, [0,1,2, 1,1,3, 2,1,4, 3,1,2]):\nGG
3 := matrix(4,3, [0,2,3, 1,2,4, 2,2,5, 3,2,3]):\nGG4 := matrix(4,3, [0
,3,0, 1,3,2, 2,3,3, 3,3,1]):\n[G1 = eval(GG1), G2 = eval(GG2), G3 = ev
al(GG3), G4 = eval(GG4)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/%#G1G-
%'matrixG6#7&7%\"\"!F+\"\"\"7%F,F+\"\"#7%F.F+\"\"$7%F0F+F,/%#G2G-F'6#7
&7%F+F,F.7%F,F,F07%F.F,\"\"%7%F0F,F./%#G3G-F'6#7&7%F+F.F07%F,F.F:7%F.F
.\"\"&7%F0F.F0/%#G4G-F'6#7&7%F+F0F+7%F,F0F.7%F.F0F07%F0F0F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "Gb1t := evalm( T &* Mb &* G
G1 ):\nGb2t := evalm( T &* Mb &* GG2 ):\nGb3t := evalm( T &* Mb &* GG3
 ):\nGb4t := evalm( T &* Mb &* GG4 ):\ntemp := transpose(evalm(T &* Mb
)):\nGb1(t) = eval(temp)^T * eval(GG1);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$Gb1G6#%\"tG*&)-%'matrixG6#7&7#,**$)F'\"\"$\"\"\"!\"
\"*$)F'\"\"#F3F2F'!\"$\"\"\"F97#,(F0F2F5!\"'F'F27#,&F0F8F5F27#F0%\"TGF
9-F+6#7&7%\"\"!FEF97%F9FEF77%F7FEF27%F2FEF9F9" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 233 "xt1 := unapply(Gb1t[1,1],t): zt1 := unapply(Gb1
t[1,3],t):\nxt2 := unapply(Gb2t[1,1],t): zt2 := unapply(Gb2t[1,3],t):
\nxt3 := unapply(Gb3t[1,1],t): zt3 := unapply(Gb3t[1,3],t):\nxt4 := un
apply(Gb4t[1,1],t): zt4 := unapply(Gb4t[1,3],t):\n\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 438 "Gbt := matrix(4,3, [ Gb1t[1,1], Gb1t[1,2
], Gb1t[1,3], \n                     Gb2t[1,1], Gb2t[1,2], Gb2t[1,3], \+
\n                     Gb3t[1,1], Gb3t[1,2], Gb3t[1,3], \n            \+
         Gb4t[1,1], Gb4t[1,2], Gb4t[1,3]]):\nGb(t) = eval(Gbt); \nplot
([[xt1(t),zt1(t),t=0..1],\n      [xt2(t),zt2(t),t=0..1],\n      [xt3(t
),zt3(t),t=0..1],\n      [xt4(t),zt4(t),t=0..1]],color=[red,green,blue
,gold],\n      thickness=4,axesfont=[HELVETICA,BOLD,14]);" }}{PARA 11 
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[l$\")Vyg>Ff[l-%+AXESLABELSG6$%!GFihm-%*AXESTICKSG6%%(DEFAULTGF]im-%%F
ONTG6%%*HELVETICAG%%BOLDG\"#9-%*THICKNESSG6#\"\"%-%%VIEWG6$F]imF]im" 
1 2 0 1 4 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "Qst := evalm( S &* Mb &* Gbt ):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "xst := collect(Qst[1,1],\{s,t\}):
\nyst := collect(Qst[1,2],\{t,s\}):\nzst := collect(Qst[1,3],\{s,t\}):
\nx(s,t) = xst; y(s,t) = yst; z(s,t) = zst;\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"xG6$%\"sG%\"tG,$F(\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6$%\"sG%\"tG,$F'\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"zG6$%\"sG%\"tG,.*&,&*$)F'\"\"$\"\"\"\"\"\"!\"$F0F0
)F(F.F/F0*&F-F/)F(\"\"#F/F1*&,&F,F.F.F0F0F(F0F0F,!\"%F'F.F0F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "xb := unapply(xst, s, t): \n
yb := unapply(yst, s, t): \nzb := unapply(zst, s, t):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "plot3d([xb(s,t), yb(s,t), zb(s,t)]
,s=0.0..1.0,t=0.0..1.0,view=[0..4,0..4,0..4],axes=BOXED,labels=[\"X\",
\"Y\",\"Z\"],axesfont=[HELVETICA,BOLD,14]);" }}{PARA 13 "" 1 "" 
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