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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 30 "Intersecting a Ray with a Cone" }}{PARA 0 "" 0 "" {TEXT 
-1 111 "Consider a cylinder with cross section in the xy plane and uni
t radius.  It has the following implicit equation" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "ceq1 := x^2+y^2=z^2: ceq1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF(F)F**$)%\"zGF(F
)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Observe that all values of \+
z are legal.  To find the intersection, solve intersect the equation f
or the ray with the equation for the cylinder." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 85 "ray1 := matrix(3,1, [x[0], y[0], z[0]]) + t * \+
matrix(3,1, [dx, dy, dz]):\nP(t) = ray1;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"PG6#%\"tG,&-%'matrixG6#7%7#&%\"xG6#\"\"!7#&%\"yGF07#&%\"zGF
0\"\"\"*&F'F8-F*6#7%7#%#dxG7#%#dyG7#%#dzGF8F8" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 154 "PS := matrix(3,1, [x[0], y[0], z[0]]):\nPD := m
atrix(3,1, [dx, dy, dz]):\nray2 := evalm(PS + scalarmul(PD,t)):\nmatri
x(3,1, [x(t), y(t), z(t)]) = eval(ray2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%'matrixG6#7%7#-%\"xG6#%\"tG7#-%\"yGF+7#-%\"zGF+-F%6#7%7#,&&F*
6#\"\"!\"\"\"*&F,F;%#dxGF;F;7#,&&F/F9F;*&F,\"\"\"%#dyGF;F;7#,&&F2F9F;*
&F,FB%#dzGF;F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Now, do the sub
stitution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 64 "ceq3 := subs(\{x=ray2[1,1],y=ray2[2,1],z=ray2[3,1]
\},ceq1):\n ceq3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$),&&%\"xG6#
\"\"!\"\"\"*&%\"tGF,%#dxGF,F,\"\"#\"\"\"F,*$),&&%\"yGF*F,*&F.F1%#dyGF,
F,F0F1F,*$),&&%\"zGF*F,*&F.F1%#dzGF,F,F0F1" }}}{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#\"\"\",$*&,**&
&%\"xG6#\"\"!F'%#dxGF'\"\"#*&&%\"yGF.F'%#dyGF'F1*&&%\"zGF.F'%#dzGF'!\"
#*$-%%sqrtG6#,4**F,\"\"\"F0FAF3FAF5FAF1**F,FAF0FAF7FAF9FAF:**F3FAF5FAF
7FAF9FAF:*&)F5F1FA)F,F1FA!\"\"*&FEFA)F7F1FAF'*&)F0F1FAFIFAF'*&FKFA)F3F
1FAFG*&)F9F1FAFFFAF'*&FOFAFMFAF'FAF1FA,(*$FEFAFG*$FKFAFG*$FOFAF'!\"\"#
F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"tG6#\"\"#,$*&,**&&%\"xG6#
\"\"!\"\"\"%#dxGF0F'*&&%\"yGF.F0%#dyGF0F'*&&%\"zGF.F0%#dzGF0!\"#*$-%%s
qrtG6#,4**F,\"\"\"F1FAF3FAF5FAF'**F,FAF1FAF7FAF9FAF:**F3FAF5FAF7FAF9FA
F:*&)F5F'FA)F,F'FA!\"\"*&FEFA)F7F'FAF0*&)F1F'FAFIFAF0*&FKFA)F3F'FAFG*&
)F9F'FAFFFAF0*&FOFAFMFAF0FAF:FA,(*$FEFAFG*$FKFAFG*$FOFAF0!\"\"#F0F'" }
}}{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*&&%\"zGF0F2%#dzGF2!\"#*$-%%sqrtG6#,4**F.\"
\"\"F7FHF:FHF<FHF8**F.FHF7FHF>FHF@FHFA**F:FHF<FHF>FHF@FHFA*&)F<F8FH)F.
F8FH!\"\"*&FLFH)F>F8FHF2*&)F7F8FHFPFHF2*&FRFH)F:F8FHFN*&)F@F8FHFMFHF2*
&FVFHFTFHF2FHF8F2F7FHFH,(*$FLFHFN*$FRFHFN*$FVFHF2!\"\"#F2F87#,&F:F2*&*
&F5FHF<FHFHFXFfnFgn7#,&F>F2*&*&F5FHF@FHFHFXFfnFgn" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"PG6#%#i2G-%'matrixG6#7%7#,&&%\"xG6#\"\"!\"\"\"*&*&
,**&F.F2%#dxGF2\"\"#*&&%\"yGF0F2%#dyGF2F8*&&%\"zGF0F2%#dzGF2!\"#*$-%%s
qrtG6#,4**F.\"\"\"F7FHF:FHF<FHF8**F.FHF7FHF>FHF@FHFA**F:FHF<FHF>FHF@FH
FA*&)F<F8FH)F.F8FH!\"\"*&FLFH)F>F8FHF2*&)F7F8FHFPFHF2*&FRFH)F:F8FHFN*&
)F@F8FHFMFHF2*&FVFHFTFHF2FHFAF2F7FHFH,(*$FLFHFN*$FRFHFN*$FVFHF2!\"\"#F
2F87#,&F:F2*&*&F5FHF<FHFHFXFfnFgn7#,&F>F2*&*&F5FHF@FHFHFXFfnFgn" }}}
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