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"" 0 "" {TEXT 239 15 "Discrete Curves" } {TEXT 239 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 31 "restar t; with(LinearAlgebra): " }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 29 "with(plots): with(plottools):" }{MPLTEXT 1 240 0 "" }}}{SECT 1 {PARA 242 "" 0 "" {TEXT 241 20 "Matrix constructio ns" }{TEXT 241 0 "" }}{EXCHG {PARA 243 "" 0 "" {TEXT 242 111 "Construc tions from the linear algebra package give a method for stacking matri ces to construct larger matrices." }{TEXT 242 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 25 " A := Matrix(3,a): " }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 35 "\n B := Matrix(3,b): " } {MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 27 "\n C :=Matrix(3,c): " }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 11 "\n A,B,C;" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 243 "" 0 "" {TEXT 242 86 "To stack the above \+ matrices side by side, use the bracket notation with vertical bars:" } {TEXT 242 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 9 " ;" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 243 "" 0 "" {TEXT 242 61 "To \+ stack them above each other, use the notation with commas." }{TEXT 242 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 6 ";" } {MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 243 "" 0 "" {TEXT 242 83 "These co nstructions are useful in connection with rotations about the x and z \+ axes." }{TEXT 242 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 78 "R1 := theta -> <<1,0,0>|<0,cos(theta),sin(theta)>|<0,-sin(theta),c os(theta)>>:" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 81 "\nR3 := theta - > <|<-sin(theta), cos(theta), 0>|<0,0,1>>:" } {MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 21 "\nR1(theta),R3(theta);" } {MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 243 77 "The func tion eucl transforms a rotation and a column vector into a 4x4 matrix" }{TEXT 243 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 35 "eucl := (A,b) -> <,<0|0|0|1>>;" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 39 "A := Matrix([[0,0,1],[1,0,0],[0,1,0]] ):" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 13 "b := <1,0,1>:" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 14 "C :=eucl(A,b);" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 232 116 "Here is a procedure to get the vector part of the 4x4 matrix. These are the first three entries in the 4th coloumn." }{TEXT 232 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 0 34 "vect := M -> M[1..3,4]: vect(C);" }{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 242 "" 0 "" {TEXT 241 25 "Graphing a discrete curve" }{TEXT 241 0 "" }}{EXCHG {PARA 243 "" 0 "" {TEXT 242 96 "This procedure plots a sequence of points as a curve by connedting consecutive points as l ines. " }{TEXT 242 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 38 "plotcurve := proc(P) local j, L, Pts; " }{MPLTEXT 1 240 0 "" } {MPLTEXT 1 240 35 "\nPts := map(x ->convert(x,list),P);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 28 "\nfor j from 2 to nops(P) do " }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 37 "\n L[j] := line(Pts[j-1],Pts[j]) od:" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 56 "\ndisplay(\{seq(L[j], j = 2. .nops(P))\}, thickness =4) end:" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 0 65 "curve1 := [<1,1,1>,<2,2,2>,<3,2,1>, <1, 1,1>]: plotcurve(curve1);" }{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 242 " " 0 "" {TEXT 241 49 "Finding the torsion angles along a discrete curve " }{TEXT 241 0 "" }}{EXCHG {PARA 243 "" 0 "" {TEXT 242 51 "Procedure t o compute torsion angle and bond anlges:" }{TEXT 242 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 50 "TorsionAngle := proc(p1,p2,p3,p 4) local x,y,a,b,c;" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 40 "\na := p 2- p1; b := p3 - p2; c:= p4 - p3;" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 54 "\na := a/Norm(a,2); b := b/Norm(b,2); c := c/Norm(c,2);" } {MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 57 "\nx := -DotProduct(a,c) + DotP roduct(a,b)*DotProduct(b,c);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 39 "\ny := DotProduct(a, CrossProduct(b,c));" }{MPLTEXT 1 240 0 "" } {MPLTEXT 1 240 36 "\nevalf((180/Pi)*argument(x+I*y), 4);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 5 "\nend:" }{MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 43 "ExtBondAngle := proc(a,b,c) lo cal v,w,x,y;" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 22 "\nv := b-a ; w \+ := c -b;" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 44 "\nx := DotProduct(v ,w)/(Norm(v,2)*Norm(w,2));" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 16 " \ny := arccos(x);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 19 "\nevalf((1 80/Pi)*y);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 5 "\nend:" }{MPLTEXT 1 240 0 "" }}}}{SECT 1 {PARA 242 "" 0 "" {TEXT 241 46 "Constructing cu rves from torsion and curvature" }{TEXT 241 0 "" }}{EXCHG {PARA 247 "" 0 "" {TEXT 244 76 "Set up construction for the basic rotation matrice s around the x and z axes." }{TEXT 244 0 "" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 245 78 "Rx := theta -> <<1,0,0>|<0,cos(theta),sin(theta )>|<0,-sin(theta),cos(theta)>>:" }{MPLTEXT 1 245 0 "" }{MPLTEXT 1 245 81 "\nRz := theta -> <|<-sin(theta), cos(thet a), 0>|<0,0,1>>:" }{MPLTEXT 1 245 0 "" }{MPLTEXT 1 245 22 "\nRx(theta) ,Rz(theta);;" }{MPLTEXT 1 245 0 "" }}}{EXCHG {PARA 247 "" 0 "" {TEXT 244 138 "Formula 13 in the notes for constructing the frames along the curve from the bond angle and torsion involves products of the folowi ng type" }{TEXT 244 0 "" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 245 74 "tr := (phi, theta, s) -> eucl(Rx(phi),<0,0,0>). eucl(Rz(theta),< s,0,0>):" }{MPLTEXT 1 245 0 "" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 245 18 "tr(phi, theta, s);" }{MPLTEXT 1 245 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 232 215 "Here is a procedure for constructing the fr ames from torsion and exterior bond angles. You can set the first tor sion angle and last two torsion angles = 0 and you can set the first a nd last exterior bond angles = 0" }{TEXT 232 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 240 61 "dff := proc(ba, ta, s) local t,j,k,N,de g,v3; N := nops(ba);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 22 "\ndeg : = evalf(Pi/180);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 27 "\nt[1] := I dentityMatrix(4);" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 22 "\nfor j fr om 2 to N do" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 49 "\n t[j] := t[j -1].tr(ta[j-1]*deg,ba[j]*deg,s[j-1])" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 5 "\n od;" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 24 "\n[seq(t[k], k = 1...N)]:" }{MPLTEXT 1 240 0 "" }{MPLTEXT 1 240 5 "\nend:" } {MPLTEXT 1 240 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 232 87 "The curv e can be obtained from the dff by finding the vector part of each 4x4 \+ matrix. " }{TEXT 232 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 0 90 "curve2 := map(x->vect(x),dff([0,70,70,70,70,70,0],[0,60,-60,60,-60 ,0,0],[1,1,1,1,1,1,1])):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 0 18 "plotcurve(curve2);" }{MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 249 "> " 0 "" {MPLTEXT 1 246 0 "" }}}{PARA 250 "" 0 "" {TEXT 236 0 "" }}{PARA 250 "" 0 "" {TEXT 236 0 "" }}{PARA 251 "" 0 "" {TEXT 222 0 "" }}{PARA 252 "" 0 "" {TEXT 212 0 "" }}{PARA 253 "" 0 "" {TEXT 247 0 "" }}{PARA 254 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }