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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 
216 "Gx := matrix(4,4, [0,1,2,3, 0,1,2,3, 0,1,2,3, 0,1,2,3]):\nGy := m
atrix(4,4, [0,0,0,0, 1,1,1,1, 2,2,2,2, 3,3,3,3]):\nGz := matrix(4,4, [
5,6,3,5, 3,5,6,3, 4,3,5,6, 5,4,3,5]):\n[Gx = eval(Gx),Gy = eval(Gy),Gz
 = eval(Gz)];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%#GxG-%'matrixG6
#7&7&\"\"!\"\"\"\"\"#\"\"$F*F*F*/%#GyG-F'6#7&7&F+F+F+F+7&F,F,F,F,7&F-F
-F-F-7&F.F.F.F./%#GzG-F'6#7&7&\"\"&\"\"'F.F>7&F.F>F?F.7&\"\"%F.F>F?7&F
>FBF.F>" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "The parametric equat
ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 195 "\nQ(s,t) = matrix(3,1, [x(s,t), y(s,t), z(s,t)]);\nx
(s,t) = S * Mb * Gx * Mb^T * T^T;\ny(s,t) = S * Mb * Gy * Mb^T * T^T;
\nz(s,t) = S * Mb * Gz * Mb^T * T^T;\n\nS = eval(S);\nT = eval(T);\nMb
 = eval(Mb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"QG6$%\"sG%\"tG-%'
matrixG6#7%7#-%\"xGF&7#-%\"yGF&7#-%\"zGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"xG6$%\"sG%\"tG*,%\"SG\"\"\"%#MbGF+%#GxGF+)F,%\"TGF
+)F/F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6$%\"sG%\"tG*,%\"SG
\"\"\"%#MbGF+%#GyGF+)F,%\"TGF+)F/F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%\"zG6$%\"sG%\"tG*,%\"SG\"\"\"%#MbGF+%#GzGF+)F,%\"TGF+)F/F/F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"SG-%'matrixG6#7#7&*$)%\"sG\"\"$\"
\"\"*$)F,\"\"#F.F,\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"TG-%'m
atrixG6#7#7&*$)%\"tG\"\"$\"\"\"*$)F,\"\"#F.F,\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%#MbG-%'matrixG6#7&7&!\"\"\"\"$!\"$\"\"\"7&F+!\"'F+
\"\"!7&F,F+F0F07&F-F0F0F0" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Pl
otting the Control Points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
249 "Pts := \{\}:\nPoints := matrix(4,4):\nfor i from 1 to 4 do\n   fo
r j from 1 to 4 do\n      temp := [Gx[i,j], Gy[i,j], Gz[i,j]]:\n      \+
Points[i,j] := matrix(3,1, temp):\n      Pts := Pts union temp:\n   od
:\nod:\nPts := convert(Pts,set):\nPoints = eval(Points);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%'PointsG-%'matrixG6#7&7&-F&6#7%7#\"\"!F-7#\"\"
&-F&6#7%7#\"\"\"F-7#\"\"'-F&6#7%7#\"\"#F-7#\"\"$-F&6#7%F=F-F/7&-F&6#7%
F-F4F=-F&6#7%F4F4F/-F&6#7%F;F4F6-F&6#7%F=F4F=7&-F&6#7%F-F;7#\"\"%-F&6#
7%F4F;F=-F&6#7%F;F;F/-F&6#7%F=F;F67&-F&6#7%F-F=F/-F&6#7%F4F=FS-F&6#7%F
;F=F=-F&6#7%F=F=F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "poin
tplot3d(Pts,color=blue,view=[0..3,0..3,3..6],\naxes=BOXED, labels=[x,y
,z], labelfont=[HELVETICA,OBLIQUE,14],\norientation=[-160,60],scaling=
CONSTRAINED,symbol=BOX);" }}{PARA 13 "" 1 "" {GLPLOT3D 242 290 290 
{PLOTDATA 3 "6*-%'POINTSG627%$\"\"#\"\"!F)$\"\"$F)7%F*F)$\"\"&F)7%F'$
\"\"\"F)$\"\"'F)7%F*F0F*7%F)F'$\"\"%F)7%F0F'F*7%F'F'F-7%F)F)F-7%F0F)F2
7%F)F0F*7%F0F0F-7%F*F'F27%F)F*F-7%F*F*F-7%F0F*F67%F'F*F*-%'SYMBOLG6#%$
BOXG-%+AXESLABELSG6&%\"xG%\"yG%\"zG-%%FONTG6%%*HELVETICAG%(OBLIQUEG\"#
9-%*AXESSTYLEGFE-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6%;F)F*Ffn;F*F2-%'C
OLOURG6&%$RGBGF)F)$\"*++++\"!\")-%+PROJECTIONG6%$!$g\"F)$\"#gF)F1" 1 
2 3 1 0 2 4 1 2 1 1.000000 60.000000 -160.000000 1 }}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "The Surface" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "MbT := transpose(Mb):\nTT \+
 := transpose(T):\nx(s,t) = S * Mb * Gx * Mb^T * T^T;\nx(s,t) = eval(S
) * eval(Mb) * eval(Gx);\neval(MbT) * eval(TT);\neval(Mb) * eval(Gx) =
 evalm(Mb &* Gx);\nevalm(Mb &* Gx) * eval(MbT) = evalm(Mb &* Gx &* MbT
);\nx(s,t) = eval(S) * evalm(Mb &* Gx &* MbT) * eval(TT);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%\"xG6$%\"sG%\"tG*,%\"SG\"\"\"%#MbGF+%#GxGF+)
F,%\"TGF+)F/F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6$%\"sG%\"t
G*(-%'matrixG6#7#7&*$)F'\"\"$\"\"\"*$)F'\"\"#F2F'\"\"\"F6-F+6#7&7&!\"
\"F1!\"$F67&F1!\"'F1\"\"!7&F<F1F?F?7&F6F?F?F?F6-F+6#7&7&F?F6F5F1FEFEFE
F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%'matrixG6#7&7&!\"\"\"\"$!\"$
\"\"\"7&F*!\"'F*\"\"!7&F+F*F/F/7&F,F/F/F/F,-F%6#7&7#*$)%\"tGF*\"\"\"7#
*$)F8\"\"#F97#F87#F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrix
G6#7&7&!\"\"\"\"$!\"$\"\"\"7&F+!\"'F+\"\"!7&F,F+F0F07&F-F0F0F0F--F&6#7
&7&F0F-\"\"#F+F6F6F6F--F&6#7&7&F0F0F0F0F;F;F6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&-%'matrixG6#7&7&\"\"!F*F*F*F)F)7&F*\"\"\"\"\"#\"\"$F
,-F&6#7&7&!\"\"F.!\"$F,7&F.!\"'F.F*7&F4F.F*F*7&F,F*F*F*F,-F&6#7&F)F)F)
7&F*F*F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6$%\"sG%\"tG*(-%'
matrixG6#7#7&*$)F'\"\"$\"\"\"*$)F'\"\"#F2F'\"\"\"F6-F+6#7&7&\"\"!F;F;F
;F:F:7&F;F;F1F;F6-F+6#7&7#*$)F(F1F27#*$)F(F5F27#F(7#F6F6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "xstm := evalm( S &* Mb &* Gx &* Mb
T &* 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 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$%\"sG%\"tG,0*&,**$)F'\"\"$\"\"\"!\"$*$)F'\"\"#
F/\"#LF'!#O\"\"*\"\"\"F7)F(F.F/F7*&,*F,!#CF1F6F'\"#F!#7F7F7)F(F3F/F7*&
,*F,\"#@F1F5F'F6F.F7F7F(F7F7F,F0F1F6F'!\"'\"\"&F7" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 79 "xb := unapply(xst, s, t): \nyb := unapply(ys
t, s, t): \nzb := unapply(zst, s, t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 197 "plot3d([xb(s,t), yb(s,t),zb(s,t)],s=0.0..1.0,t=0.0..1.0,view=[0
..3,0..3,3.5..5.5],\naxes=BOXED, labels=[x,y,z], labelfont=[HELVETICA,
BOLD,14],\norientation=[-160,60],scaling=CONSTRAINED,style=patch);" }}
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