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}
{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Bezier Surfaces." }}
{PARA 0 "" 0 "" {TEXT -1 71 "This worksheet developes a quite simple f
ormulation of a Bezier Surface" }}{PARA 0 "" 0 "" {TEXT -1 42 "control
led by three 4x4 geometry matrices." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 105 "                                       \+
                               J. Ross Beveridge           4/27/99" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "I
nitialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(linalg
):\nwith(plots):\nreadlib(unassign):" }}}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 50 "Defining the x, y and z parameterized expressions." }}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 55 "The basics, the two parameter vec
tors and basis matrix." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "S
 := 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]):\n" 
}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "The Geometry Matrices" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
234 "Gx := matrix(4,4, [2.5,1,4,2.5, 2,1,4,3, 2,1,4,3, 2.5,1,4,2.5]):
\nGy := matrix(4,4, [2.5,1,1,2.5, 2,1,1,2, 3,4,4,3, 2.5,4,4,2.5]):\nGz
 := matrix(4,4, [0,1,1,0, 1,2,2,1, 1,2,2,1, 0,1,1,0]):\n[Gx = eval(Gx)
, Gy = eval(Gy), Gz = eval(Gz)];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7%/%#GxG-%'matrixG6#7&7&$\"#D!\"\"\"\"\"\"\"%F+7&\"\"#F.F/\"\"$F0F*/%#
GyG-F'6#7&7&F+F.F.F+7&F1F.F.F17&F2F/F/F27&F+F/F/F+/%#GzG-F'6#7&7&\"\"!
F.F.FB7&F.F1F1F.FCFA" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Plottin
g the Control Points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "Pts
 := \{\}:\nPoints := matrix(4,4):\nfor i from 1 to 4 do\n   for j from
 1 to 4 do\n      temp := [Gx[i,j], Gy[i,j], Gz[i,j]]:\n      Points[i
,j] := matrix(3,1, temp):\n      Pts := Pts union temp:\n   od:\nod:\n
Pts := convert(Pts,set):\nPoints = eval(Points);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 141 "pointplot3d(Pts,color=blue,\naxes=BOXED, labels=[x,y
,z], labelfont=[HELVETICA,BOLD,14],\norientation=[-160,60],scaling=CON
STRAINED,symbol=BOX);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'PointsG-%'
matrixG6#7&7&-F&6#7%7#$\"#D!\"\"F-7#\"\"!-F&6#7%7#\"\"\"F6F6-F&6#7%7#
\"\"%F6F6F*7&-F&6#7%7#\"\"#FAF6-F&6#7%F6F6FA-F&6#7%F;F6FA-F&6#7%7#\"\"
$FAF67&-F&6#7%FAFLF6-F&6#7%F6F;FA-F&6#7%F;F;FA-F&6#7%FLFLF67&F*-F&6#7%
F6F;F6-F&6#7%F;F;F6F*" }}{PARA 13 "" 1 "" {GLPLOT3D 242 290 290 
{PLOTDATA 3 "6)-%'POINTSG6/7%$\"#D!\"\"F'\"\"!7%$\"\"%F*F,$\"\"\"F*7%F
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.F.7%F,F,F27%F7F7F.7%F.F,F.-%'SYMBOLG6#%$BOXG-%+AXESLABELSG6&%\"xG%\"y
G%\"zG-%%FONTG6%%*HELVETICAG%%BOLDG\"#9-%*AXESSTYLEGFA-%(SCALINGG6#%,C
ONSTRAINEDG-%'COLOURG6&%$RGBGF*F*$\"*++++\"!\")-%+PROJECTIONG6%$!$g\"F
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "
The Surface" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "\nMbT := tra
nspose(Mb):\nTT  := transpose(T):\nxstm := evalm( S &* Mb &* Gx &* MbT
 &* TT):\nystm := evalm( S &* Mb &* Gy &* MbT &* TT):\nzstm := evalm( \+
S &* Mb &* Gz &* MbT &* TT):\nxst := collect(xstm[1,1],\{s,t\}):\nyst \+
:= collect(ystm[1,1],\{s,t\}):\nzst := collect(zstm[1,1],\{s,t\}):\nx(
s,t) = xst; y(s,t) = yst; z(s,t) = zst;\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "xb := unapply(xst, s, t): \nyb := unapply(yst, s, t):
 \nzb := unapply(zst, s, t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "pl
ot3d([xb(s,t), yb(s,t),zb(s,t)],s=0.0..1.0,t=0.0..1.0,\naxes=BOXED, la
bels=[x,y,z], labelfont=[HELVETICA,BOLD,14],\norientation=[-160,60],sc
aling=CONSTRAINED,style=patch);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"xG6$%\"sG%\"tG,.*&,(*$)F'\"\"#\"\"\"$!#I!\"\"F'$\"#IF2$!#!*F2\"\"\"F
7)F(\"\"$F/F7*&,(F,$\"#XF2F'$!#XF2$\"$N\"F2F7F7)F(F.F/F7*&,(F,F>F'F<F>
F7F7F(F7F7F,$\"#:F2F'$!#:F2$\"#DF2F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%\"yG6$%\"sG%\"tG,.$\"#D!\"\"\"\"\"*&,**$)F'\"\"$\"\"\"$\"\"*\"\"!
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F,F'F>F<F-F-F(F-F-F0$!\"$F6F7F>F'$!#:F," }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"zG6$%\"sG%\"tG,**$)F(\"\"#\"\"\"!\"$F(\"\"$*$)F'F,F-F.F'F/
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