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Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 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 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 70 " This work sheet will plot the (T,N,B) coordinate system on a given" }}{PARA 0 "" 0 "" {TEXT 212 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 212 0 "" }} {PARA 0 "" 0 "" {TEXT 212 71 " Define r. I have put in an r. Just ch ange the components for your r." }}{PARA 0 "" 0 "" {TEXT 212 19 " My r is a helix." }}{PARA 0 "" 0 "" {TEXT 212 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 := unapply(r,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 9 " \+ Plot r" }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "spacec urve(\{r,rf(t)\},t=0..4*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 77 " Determine T,N and B by taking the appropriate derivat ives and normalizing." }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 81 " Create the velocity and ac celeration vectors from r. Determine the lengths of " }}{PARA 0 "" 0 " " {TEXT 212 17 " these vectors." }}{PARA 0 "" 0 "" {TEXT 212 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 39 "dsdt := simplify(sqrt(innerprod(v,v)));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a := diff(v,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "af := unapply(a,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 2 " " }}{PARA 0 "" 0 "" {TEXT 208 67 " Before we calculate \{T,N,B\}, we give the cu rvature of this curve." }}{PARA 0 "" 0 "" {TEXT 208 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 0 1 "numerator := simplify(crossprod(v, a));" "-I%mrowG6#/I+ modulenameG6\"I,TypesettingGI(_syslibGF'6)-I#miGF$6&Q*numeratorF'/%'it alicGQ%trueF'/%,mathvariantGQ,bold-italicF'/%+fontweightGQ%boldF'-I#mo GF$6.Q\"~F'/F3F7F5/%&fenceGQ&falseF'/%*separatorGF?/%)stretchyGF?/%*sy mmetricGF?/%(largeopGF?/%.movablelimitsGF?/%'accentGF?/%'lspaceGQ&0.0e mF'/%'rspaceGFN-F96.Q*≔F'F " 0 "" {XPPEDIT 0 1 "curvature := simplify(`/`(`*`(sqrt(linalg:-innerprod(numerator, numer ator))), `*`(`^`(dsdt, 3))));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6)-I#miGF$6&Q*curvatureF'/%'italicGQ%trueF'/%,mathvaria ntGQ,bold-italicF'/%+fontweightGQ%boldF'-I#moGF$6.Q\"~F'/F3F7F5/%&fenc eGQ&falseF'/%*separatorGF?/%)stretchyGF?/%*symmetricGF?/%(largeopGF?/% .movablelimitsGF?/%'accentGF?/%'lspaceGQ&0.0emF'/%'rspaceGFN-F96.Q*&co loneq;F'F " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 79 " Create T by making it a unit vector in the direction of the velocit y vector." }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {XPPEDIT 0 1 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "T := simplify([v[1]/dsdt,v[2]/dsdt,v[3]/dsdt]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }} {PARA 0 "" 0 "" {TEXT 212 63 " Create N by making it a unit vector \+ in the direction of T'." }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 0 1 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Tp := simplify(diff(T,t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "TpM := simplify(sqrt(innerprod(Tp,Tp)));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "N := simplify([Tp[1]/TpM,Tp[ 2]/TpM,Tp[3]/TpM]);" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 0 1 "" "%#%? G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 52 " Create B by taking the cross product of T and N." }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "B := simplify(crossprod( T,N));" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "B := [B[1], B[2], B [3]];" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6(-I#miGF $6%Q\"BF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"~F'/F 3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetric GF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%'lspaceGQ&0.0emF'/%'r spaceGFL-F66-Q*≔F'F9F;F>F@FBFDFFFH/FKQ,0.2777778emF'/FNFSF5-I( mfencedGF$6&-F#6*F+-FV6&-F#6#-I#mnGF$6$Q\"1F'F9F9/%%openGQ\"[F'/%&clos eGQ\"]F'-F66-Q\",F'F9F;/F?F1F@FBFDFFFHFJ/FNQ,0.3333333emF'F+-FV6&-F#6# -Fin6$Q\"2F'F9F9F\\oF_oFboF+-FV6&-F#6#-Fin6$Q\"3F'F9F9F\\oF_oF9F\\oF_o -F66-Q\";F'F9F;FeoF@FBFDFFFHFJFT" }}}{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 19 "Bf := unapply(B,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 60 " Check to make sure T,N and B are unit orthogonal vectors" }} {PARA 0 "" 0 "" {TEXT 212 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 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "c := simplify(crossprod( T,B));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 69 " t0 is the \+ starting t value, ti is the time increment value between" }}{PARA 0 "" 0 "" {TEXT 212 20 " plots of (T,N,B)." }}{PARA 0 "" 0 "" {TEXT 212 49 " number_of_TNB is how many plots starting at t0" }}{PARA 0 "" 0 "" {TEXT 212 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 212 1 " " }}{PARA 0 "" 0 "" {TEXT 212 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 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 91 " Gene rate the plots for r, T, N, B and a. This is done to create an animati on showing the" }}{PARA 0 "" 0 "" {TEXT 212 28 " (T,N,B) system on y our r." }}{PARA 0 "" 0 "" {TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "g := sp acecurve(r,t=t0..number_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 0 124 " g2[k] := s pacecurve(\{[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=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 " g3[k] := spacecurve(\{[rf(k*ti)[1]+t*Nf(k*ti)[1],r f(k*ti)[2]+t*Nf(k*ti)[2],rf(k*ti)[3]+t*Nf(k*ti)[3]]\},t=0..tN,color=gr een):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 " 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,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 " 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=yellow):" }}{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 27 " pic[k] := display(g,g1[k]" }{MPLTEXT 1 0 6 ",g2[k]" }{MPLTEXT 1 0 6 ",g3[k]" }{MPLTEXT 1 0 6 ",g4[k]" }{MPLTEXT 1 0 6 ",g 5[k]" }{MPLTEXT 1 0 2 "):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT 212 91 " The next command cre ates an animation of your curve and the (T,N,B) system. Click on the" }}{PARA 0 "" 0 "" {TEXT 212 40 " picture and then hit the play butto n." }}{PARA 0 "" 0 "" {TEXT 212 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=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 2 1 1805 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }