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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 34 "Intersecting a Ray with a Cylinder" }}{PARA 0 "" 0 "" 
{TEXT -1 111 "Consider a cylinder with cross section in the xy plane a
nd unit radius.  It has the following implicit equation" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ceq1 := x^2+y^2=1: ceq1;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF(F)F*F*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Observe that all values of z
 are legal.  To find the intersection, solve intersect the equation fo
r the ray with the equation for the cylinder." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 85 "ray1 := matrix(3,1, [x[0], y[0], z[0]]) + t * ma
trix(3,1, [dx, dy, dz]):\nP(t) = ray1;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"PG6#%\"tG,&-%'matrixG6#7%7#&%\"xG6#\"\"!7#&%\"yGF07#&%\"zGF
0\"\"\"*&F'F8-F*6#7%7#%#dxG7#%#dyG7#%#dzGF8F8" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 154 "PS := matrix(3,1, [x[0], y[0], z[0]]):\nPD := m
atrix(3,1, [dx, dy, dz]):\nray2 := evalm(PS + scalarmul(PD,t)):\nmatri
x(3,1, [x(t), y(t), z(t)]) = eval(ray2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%'matrixG6#7%7#-%\"xG6#%\"tG7#-%\"yGF+7#-%\"zGF+-F%6#7%7#,&&F*
6#\"\"!\"\"\"*&F,F;%#dxGF;F;7#,&&F/F9F;*&F,\"\"\"%#dyGF;F;7#,&&F2F9F;*
&F,FB%#dzGF;F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Now, do the sub
stitution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "ceq3 := subs(\{x=ray2[1,1],y=ray2[2,1]\},ceq1):\n c
eq3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$),&&%\"xG6#\"\"!\"\"\"*&%
\"tGF,%#dxGF,F,\"\"#\"\"\"F,*$),&&%\"yGF*F,*&F.F1%#dyGF,F,F0F1F,F," }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "troots := solve(ceq3,t):\nt
[1] = troots[1];\nt[2] = troots[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/&%\"tG6#\"\"\",$*&,(*&&%\"xG6#\"\"!F'%#dxGF'!\"#*&&%\"yGF.F'%#dyGF'F
1*$-%%sqrtG6#,,**F,\"\"\"F0F<F3F<F5F<\"\"#*&)F5F=F<)F,F=F<!\"\"*$F?F<F
'*$)F0F=F<F'*&FDF<)F3F=F<FAF<F=F<,&FBF'FCF'!\"\"#F'F=" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/&%\"tG6#\"\"#,$*&,(*&&%\"xG6#\"\"!\"\"\"%#dxGF0!
\"#*&&%\"yGF.F0%#dyGF0F2*$-%%sqrtG6#,,**F,\"\"\"F1F=F4F=F6F=F'*&)F6F'F
=)F,F'F=!\"\"*$F?F=F0*$)F1F'F=F0*&FDF=)F4F'F=FAF=F2F=,&FBF0FCF0!\"\"#F
0F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "This will yield to poitns \+
in 3D" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "P[i1] = subs(\{t=troots[1]\},eval(ray2));\nP[i2] = su
bs(\{t=troots[2]\},eval(ray2));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
&%\"PG6#%#i1G-%'matrixG6#7%7#,&&%\"xG6#\"\"!\"\"\"*&*&,(*&F.F2%#dxGF2!
\"#*&&%\"yGF0F2%#dyGF2F8*$-%%sqrtG6#,,**F.\"\"\"F7FCF:FCF<FC\"\"#*&)F<
FDFC)F.FDFC!\"\"*$FFFCF2*$)F7FDFCF2*&FKFC)F:FDFCFHFCFDF2F7FCFC,&FIF2FJ
F2!\"\"#F2FD7#,&F:F2*&*&F5FCF<FCFCFNFOFP7#,&&%\"zGF0F2*&*&F5FC%#dzGF2F
CFNFOFP" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"PG6#%#i2G-%'matrixG6#7
%7#,&&%\"xG6#\"\"!\"\"\"*&*&,(*&F.F2%#dxGF2!\"#*&&%\"yGF0F2%#dyGF2F8*$
-%%sqrtG6#,,**F.\"\"\"F7FCF:FCF<FC\"\"#*&)F<FDFC)F.FDFC!\"\"*$FFFCF2*$
)F7FDFCF2*&FKFC)F:FDFCFHFCF8F2F7FCFC,&FIF2FJF2!\"\"#F2FD7#,&F:F2*&*&F5
FCF<FCFCFNFOFP7#,&&%\"zGF0F2*&*&F5FC%#dzGF2FCFNFOFP" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "2 11 0 0" 0 }{VIEWOPTS 1 1 
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