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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 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):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning,
 new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new \+
definition for trace" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Hermite
 Curves" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 "The Basis Matrix and P
arameter 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-%'matrix
G6#7&7&\"\"#!\"#\"\"\"F,7&!\"$\"\"$F+!\"\"7&\"\"!F2F,F27&F,F2F2F2" }}}
}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 57 "Define and Show the four curves \+
that control the surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
295 "GG1 := matrix(4,3, [ 1, 0, 0,   1,  0,  1,   0,  0, 2,   0,  0, -
2]):\nGG2 := matrix(4,3, [ 0, 1, 0,   0,  1,  1,   0,  0, 2,   0,  0, \+
-2]):\nGG3 := matrix(4,3, [-3, 0, 0,  -3,  0,  0.1,   0,  0, 0.05,   0
,  0, -0.05]):\nGG4 := matrix(4,3, [ 0, 3, 0,   0,  3,  0.1,   0,  0, \+
0.05,   0,  0, -0.05]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 490 "Gt1 := \+
evalm( 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 \+
= eval(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\"\"\"
%\"MGF+%#G1GF+/-%#P4GF'*(F*\"\"\"F,F2%#G2GF+/-%#R1GF'*(F*F2F,F2%#G3GF+
/-%#R4GF'*(F*F2F,F2%#G4GF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/%#G1G
-%'matrixG6#7&7%\"\"\"\"\"!F,7%F+F,F+7%F,F,\"\"#7%F,F,!\"#/%#G2G-F'6#7
&7%F,F+F,7%F,F+F+F.F0/%#G3G-F'6#7&7%!\"$F,F,7%F?F,$F+!\"\"7%F,F,$\"\"&
F17%F,F,$!\"&F1/%#G4G-F'6#7&7%F,\"\"$F,7%F,FOFAFCFF" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7%/-%\"xG6$%\"sG%\"tG*(%\"SG\"\"\"%\"MGF,-%'matrixG6#
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,/-%\"z
GF'*(F+FHF-FH-F/6#7&7#-&F56#FenF77#-&F;F]oF77#-&F?F]oF77#-&FCF]oF7F," 
}}}{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 := unappl
y(Gt2[1,1],t): y2 := unapply(Gt2[1,2],t): z2 := unapply(Gt2[1,3],t):\n
x3 := unapply(Gt3[1,1],t): y3 := unapply(Gt3[1,2],t): z3 := unapply(Gt
3[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 "sample :
= 100:\nPts1 := [seq([x1(i/sample), y1(i/sample), z1(i/sample)],i=0..s
ample)]:\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/sample)],
i=0..sample)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "PLOT3D(C
URVES(eval(Pts1),COLOR(RGB,1.0,0.0,0.0)),\n       CURVES(eval(Pts2),CO
LOR(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       TITLE(C
URVES),AXES(BOXED),LABELS([x,y,z]),FONT(HELVETICA,BOLD,14),THICKNESS(4
));" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
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m7%F)F]imF]gm7%F)F]imF`gm7%F)F]imFcgm7%F)F]imFfgm7%F)F]imFigm7%F)F]imF
\\hm7%F)F]imF_hm7%F)F]imFbhm7%F)F]imFehm-Fc^l6&Fe^lF)Ff^lFf^l-%&TITLEG
6#F$-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%%\"xG%\"yG%\"zG-%%FONTG6%%*HE
LVETICAG%%BOLDG\"#9-%*THICKNESSG6#\"\"%" 1 2 0 1 4 2 1 1 2 2 1.000000 
35.000000 42.000000 0 }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 68 "Now co
mpose the four t dependent geometries into the Hermite Surface" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
536 "GMH := matrix(4,3, [Gt1[1,1], Gt1[1,2], Gt1[1,3], \n             \+
       Gt2[1,1], Gt2[1,2], Gt2[1,3], \n                    Gt3[1,1], G
t3[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(colle
ct(QSH[1,2],\{t,s\}), t,s):\nzh := unapply(collect(QSH[1,3],\{t,s\}), \+
t,s):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"QG6$%\"sG%\"tG*(%\"SG
\"\"\"%\"MGF+-%\"GG6#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"QG6$
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{XPPMATH 20 "6#/-%\"yG6$%\"sG%\"tG*$)F'\"\"$\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"zG6$%\"sG%\"tG,(*&,**$)F'\"\"$\"\"\"$!#S!\"#*$)F
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