{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " This work sheet will plot t he (T,N,B) coordinate system on a given" }}{PARA 0 "" 0 "" {TEXT -1 82 " curve r(t)=[x(t),y(t),z(t)] for t0 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 71 " Define r. I have put in an r. Just change the c omponents for your r." }}{PARA 0 "" 0 "" {TEXT -1 19 " My r is a hel ix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r := [t ,sin(t),cos(t)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rf := u napply(r,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " Plot r " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "spacecu rve(\{r,rf(t)\},t=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 " Determine T,N and B by taking the appropriate derivatives and n ormalizing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 " Create the velocity and acceleration vectors from r. Determine the lengths of " }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ these vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " v := diff(r,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a := dif f(v,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "af := unapply(a, t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lv := sqrt(innerprod (v,v));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 " Create \+ T by making it a unit vector in the direction of the velocity vector. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "T := [v[1]/lv,v[2]/lv,v[3]/lv];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "u := diff(T,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lu := sqrt(innerprod(u,u));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 63 " Create N by making it a unit vector i n the direction of T'." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N := [u[1]/lu,u[2]/lu,u[3]/lu];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 " Create B by taking the cross product o f T and N." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B := crossprod(T,N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplif y(B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Tf := unapply(T,t) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Nf := unapply(N,t);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B := [B[1],B[2],B[3]];" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Bf := unapply(B,t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " Check to make sure T, N and B are unit orthogonal vectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(T,N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(B,N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(T,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(T,T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(N,N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "innerprod(B,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "b := si mplify(crossprod(T,N));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " simplify(B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c := simpli fy(crossprod(T,B));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simp lify(N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " t0 is \+ the starting t value, ti is the time increment value between" }}{PARA 0 "" 0 "" {TEXT -1 20 " plots of (T,N,B)." }}{PARA 0 "" 0 "" {TEXT -1 49 " number_of_TNB is how many plots starting at t0" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "t0 := 0;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "ti := Pi/4.;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "number_of_TNB := 20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 58 " Each of T,N,B is plotted as a line for 0 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "tT := 1.;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tN := 1.;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "tB := 1.;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ta := 1.;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 " Generate the plots for r, T, N, B and a. This is done to create an animation showing the" }} {PARA 0 "" 0 "" {TEXT -1 28 " (T,N,B) system on your r." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "g := spacecurve(r,t=t0..numb er_of_TNB*ti+t0/4,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for k from 1 by 1 to number_of_TNB do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " g1[k] := spacecurve(r,t=t0..k*ti+t0/4,color=black): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 " g2[k] := spacecurve(\{[rf(k* ti)[1]+t*Tf(k*ti)[1],rf(k*ti)[2]+t*Tf(k*ti)[2],rf(k*ti)[3]+t*Tf(k*ti)[ 3]]\},t=0..tT,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 " g3[ k] := spacecurve(\{[rf(k*ti)[1]+t*Nf(k*ti)[1],rf(k*ti)[2]+t*Nf(k*ti)[2 ],rf(k*ti)[3]+t*Nf(k*ti)[3]]\},t=0..tN,color=green):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 " g4[k] := spacecurve(\{[rf(k*ti)[1]+t*Bf(k*ti) [1],rf(k*ti)[2]+t*Bf(k*ti)[2],rf(k*ti)[3]+t*Bf(k*ti)[3]]\},t=0..tB,col or=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 " g5[k] := spacecurve (\{[rf(k*ti)[1]+t*af(k*ti)[1],rf(k*ti)[2]+t*af(k*ti)[2],rf(k*ti)[3]+t* af(k*ti)[3]]\},t=0..ta,color=black):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for k from 1 by 1 to number_of_TNB do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " pic[k] \+ := display(g,g1[k],g2[k],g3[k],g4[k],g5[k]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 " \+ The next command creates an animation of your curve and the (T,N,B) s ystem. Click on the" }}{PARA 0 "" 0 "" {TEXT -1 40 " picture and the n hit the play button." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "display([seq(pic[i],i=1..number_of_TNB)],insequence=t rue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }