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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 1 {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 
216 "Gx := matrix(4,4, [2,1,4,3, 2,1,4,3, 2,1,4,3, 2,1,4,3]):\nGy := m
atrix(4,4, [2,1,1,2, 2,1,1,2, 3,4,4,3, 3,4,4,3]):\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&\"\"#\"\"\"\"\"%\"\"$F*F*F*/%#GyG-F'6#7&7&F+F,F,F+F47&F.F-F-F.F5/
%#GzG-F'6#7&7&\"\"!F,F,F<7&F,F+F+F,F=F;" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 27 "Plotting the Control Points" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 249 "Pts := \{\}:\nPoints := matrix(4,4):\nfor i from 1 t
o 4 do\n   for j from 1 to 4 do\n      temp := [Gx[i,j], Gy[i,j], Gz[i
,j]]:\n      Points[i,j] := matrix(3,1, temp):\n      Pts := Pts union
 temp:\n   od:\nod:\nPts := convert(Pts,set):\nPoints = eval(Points);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "pointplot3d(Pts,color=blue,\na
xes=BOXED, labels=[x,y,z], labelfont=[HELVETICA,BOLD,14],\norientation
=[-160,60],scaling=CONSTRAINED,symbol=BOX);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%'PointsG-%'matrixG6#7&7&-F&6#7%7#\"\"#F-7#\"\"!-F&6#7
%7#\"\"\"F4F4-F&6#7%7#\"\"%F4F4-F&6#7%7#\"\"$F-F/7&-F&6#7%F-F-F4-F&6#7
%F4F4F--F&6#7%F9F4F--F&6#7%F>F-F47&-F&6#7%F-F>F4-F&6#7%F4F9F--F&6#7%F9
F9F--F&6#7%F>F>F47&-F&6#7%F-F>F/-F&6#7%F4F9F4-F&6#7%F9F9F4-F&6#7%F>F>F
/" }}{PARA 13 "" 1 "" {GLPLOT3D 436 289 289 {PLOTDATA 3 "6)-%'POINTSG6
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%F'F2F*7%F.F*F)7%F2F2F*7%F*F*F)7%F.F.F'7%F*F.F)7%F'F2F'7%F2F2F'7%F.F.F
)7%F'F'F'7%F2F'F'-%'SYMBOLG6#%$BOXG-%+AXESLABELSG6&%\"xG%\"yG%\"zG-%%F
ONTG6%%*HELVETICAG%%BOLDG\"#9-%*AXESSTYLEGFA-%(SCALINGG6#%,CONSTRAINED
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0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "The Surface
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "\nMbT := transpose(Mb):
\nTT  := transpose(T):\nxstm := evalm( S &* Mb &* Gx &* MbT &* TT):\ny
stm := evalm( S &* Mb &* Gy &* MbT &* TT):\nzstm := evalm( S &* Mb &* \+
Gz &* MbT &* TT):\nxst := collect(xstm[1,1],\{s,t\}):\nyst := collect(
ystm[1,1],\{s,t\}):\nzst := collect(zstm[1,1],\{s,t\}):\nx(s,t) = xst;
 y(s,t) = yst; z(s,t) = zst;\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x
b := unapply(xst, s, t): \nyb := unapply(yst, s, t): \nzb := unapply(z
st, s, t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "plot3d([xb(s,t), yb(
s,t),zb(s,t)],s=0.0..1.0,t=0.0..1.0,\naxes=BOXED, labels=[x,y,z], labe
lfont=[HELVETICA,OBLIQUE,14],\norientation=[-160,60],scaling=CONSTRAIN
ED,style=patch);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6$%\"sG%\"t
G,**$)F(\"\"$\"\"\"!\")*$)F(\"\"#F-\"#7F(!\"$F1\"\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"yG6$%\"sG%\"tG,,\"\"#\"\"\"*&,(*$)F'\"\"$\"\"
\"\"#7*$)F'F*F1!#=F0F+F+)F(F*F1F+*&,(F.!#7F3\"#=!\"$F+F+F(F+F+F.!\"#F3
F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"zG6$%\"sG%\"tG,**$)F(\"\"#
\"\"\"!\"$F(\"\"$*$)F'F,F-F.F'F/" }}{PARA 13 "" 1 "" {GLPLOT3D 505 
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