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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 4530" }}{PARA 257 "" 0 "" 
{TEXT -1 31 "Computing curvature and torsion" }}{PARA 258 "" 0 "" 
{TEXT -1 26 "Wednesday, January 26 2005" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 236 "These procedures are like those in \+
our text, but may not be exactly the same....(Maple has added subrouti
nes for dot and cross product, for example, so that you don't need to \+
write your own like I did in this file from three years ago.)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "dp:=proc(X,Y
)  #dotproduct\nX[1]*Y[1]+X[2]*Y[2]+X[3]*Y[3];\nend:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "x:=[a,b,c]:  #example\ny:=[r,s,t]:\ndp(x,
y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"aG\"\"\"%\"rGF&F&*&%\"bG
F&%\"sGF&F&*&%\"cGF&%\"tGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 "nrm:=proc(X)  #magnitude (norm)\nsqrt(dp(X,X));\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "nrm(y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$,(*$)%\"rG\"\"#\"\"\"F)*$)%\"sGF(F)F)*$)%\"tGF(F)F)#F
)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "xp:=proc(X,Y)  #cro
ss product\nlocal a,b,c;\na:=X[2]*Y[3]-X[3]*Y[2];\nb:=X[3]*Y[1]-Y[3]*X
[1];\nc:=X[1]*Y[2]-X[2]*Y[1];\n[a,b,c];\nend:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "xp(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*
&%\"bG\"\"\"%\"tGF'F'*&%\"cGF'%\"sGF'!\"\",&*&F*F'%\"rGF'F'*&F(F'%\"aG
F'F,,&*&F1F'F+F'F'*&F&F'F/F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 229 "curv:=proc(alpha)   #curvature\nlocal alphap,alphapp,num,deno
m;\nalphap:=diff(alpha,t);\nalphapp:=diff(alphap,t);\nnum:=simplify(nr
m(xp(alphap,alphapp)),radical,symbolic);\ndenom:=nrm(alphap)^3;\nRETUR
N(kappa=simplify(num/denom));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 222 "tor:=proc(alpha)\nlocal alphap,alphapp,alphappp,num;
\nalphap:=diff(alpha,t);\nalphapp:=diff(alphap,t);\nalphappp:=diff(alp
happ,t);\nnum:=dp(xp(alphap,alphapp),alphappp);\nRETURN(tau=simplify(n
um/nrm(xp(alphap,alphapp))^2));\nend:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 22 "Helicoid, we just did:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "hel:=[a*cos(t),a*sin(t),b*t];\ncurv(hel);tor(hel);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$helG7%*&%\"aG\"\"\"-%$cosG6#%\"tGF
(*&F'F(-%$sinGF+F(*&%\"bGF(F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&
kappaG*&%\"aG\"\"\",&*$)%\"bG\"\"#F'F'*$)F&F,F'F'!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%$tauG*&%\"bG\"\"\",&*$)F&\"\"#F'F'*$)%\"aGF+F'F'
!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "17 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }