{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "G:=proc(surf,u,v)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " local f1,f2,g11,g12,g22;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " f1:=(u,v)->subs(uu=u,diff(surf(uu,v),uu));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " f2:=(u,v)->subs(vv=v,diff(surf(u,vv),vv) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " g11:=simplify(dotprod(f1(u,v ),f1(u,v),orthogonal));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " g12:=si mplify(dotprod(f1(u,v),f2(u,v),orthogonal));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " g22:=simplify(dotprod(f2(u,v),f2(u,v),orthogonal)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " [g11,g12,g22];" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Christoffel:=proc(surf,u,v)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 " local _G,detG,diffG_u,diffG_v,gamma_111,gamma_112,gamma_121,gamma_122 ,gamma_221,gamma_222;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " _G:=evalm (G(surf,uu,vv));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " detG:=det([[_G [1],_G[2]],[_G[2],_G[3]]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " dif fG_u:=[diff(_G[1],uu),diff(_G[2],uu),diff(_G[3],uu)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " diffG_v:=[diff(_G[1],vv),diff(_G[2],vv),diff(_G [3],vv)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 " gamma_111:=simplify(( _G[3]*diffG_u[1]-2*_G[2]*diffG_u[2]+_G[2]*diffG_v[1])/(2*detG));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " gamma_112:=simplify((-_G[2]*diffG_ u[1]+2*_G[1]*diffG_u[2]-_G[1]*diffG_v[1])/(2*detG));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " gamma_121:=simplify((_G[3]*diffG_v[1]-_G[2]*dif fG_u[3])/(2*detG));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " gamma_122:= simplify((_G[1]*diffG_u[3]-_G[2]*diffG_v[1])/(2*detG));" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 87 " gamma_221:=simplify((-_G[2]*diffG_v[3]+2*_G[3 ]*diffG_v[2]-_G[3]*diffG_u[3])/(2*detG));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 " gamma_222:=simplify((_G[1]*diffG_v[3]-2*_G[2]*diffG_ v[2]+_G[2]*diffG_u[3])/(2*detG));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 " subs(uu=u,vv=v,[gamma_111,gamma_112,gamma_121,gamma_122,gamma_221 ,gamma_222]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "helicoid:=(u,v)->[v*cos(u),v*sin(u) ,u];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Christoffel(helicoi d,u[1],u[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "geodesic_ eq:=proc(f,u,v) \n local eq1,eq2;\n eq1:=diff(u(t),t,t)+Christoffel( f,u(t),v(t))[1]*diff(u(t),t)^2\n +2*Christoffel(f,u(t),v(t))[3 ]*diff(u(t),t)*diff(v(t),t)\n +Christoffel(f,u(t),v (t))[5]*diff(v(t),t)^2=0;\n eq2:=diff(v(t),t,t)+Christoffel(f,u(t),v(t ))[2]*diff(u(t),t)^2\n +2*Christoffel(f,u(t),v(t))[4]*diff(u(t ),t)*diff(v(t),t)\n +Christoffel(f,u(t),v(t))[6]*di ff(v(t),t)^2=0;\n eq1,eq2; \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "geodesic_eq(helicoid,u[1],u[2]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 47 "with(plots):setoptions3d(scaling=constrained ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "geodesic_plot:=proc (f,u0,v0,Du0,Dv0,ustart,uend,vstart,vend,tstart,tend)\n local sys,desy s,u1,v1,G0,pom,GEO,SURF;\n sys:=geodesic_eq(f,u,v);\n G0:=G(f,u0,v0); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " pom:=evalm([Du0,Dv0]&*[[G0[1], G0[2]],[G0[2],G0[3]]]&*[[Du0],[Dv0]]); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " desys:=dsolve(\{sys,u(0)=u0,v(0)=v0,D(u)(0)=Du0/sqrt (pom[1]),D(v)(0)=Dv0/sqrt(pom[1])\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " \{u(t),v(t)\},type=numeric,output=listprocedure); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 237 " u1:=subs(desys,u(t)); v1:=sub s(desys,v(t));\n GEO:=spacecurve(subs(\{u='u1'(t),v='v1'(t)\},f(u,v)), t=tstart..tend,color=red,thickness=3):\n SURF:=plot3d(subs(\{u=u(t),v= v(t)\},f(u,v)),u=ustart..uend,v=vstart..vend,color=grey,style=wirefram e):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " display(GEO,SURF):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "geodesics_plot:=proc(f,u0,v0,ustart,uend,vstart,vend ,tstart,tend,n)\n local i,sys,desys1,desys2,u1,v1,u2,v2,G0,pom,b1,b2,s ol1,sol2,sol,GEO1,GEO2,SURF,POINT;\n sys:=geodesic_eq(f,u,v);G0:=G(f,u 0,v0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " pom:=evalm([1,1]&*[[G0[1 ],G0[2]],[G0[2],G0[3]]]&*[[1],[1]]); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " for i from 0 to iquo(n,2)+1 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " b1:='b1':b2:=(cos(i*2*Pi/n)*sqrt(pom[1])-(G0[1]+G0[ 2])*b1)/(G0[2]+G0[3]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 " sol:=so lve(evalm([b1,b2]&*[[G0[1],G0[2]],[G0[2],G0[3]]]&*[[b1],[b2]])[1]=1,b1 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 " sol1:=sol[1];sol2:=sol[2]; desys1:=dsolve(\{sys,u(0)=u0,v(0)=v0,D(u)(0)=sol1,D(v)(0)=su bs(b1=sol1,b2)\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 " \+ \{u(t),v(t)\},type=numeric,output=listprocedure);\n \+ desys2:=dsolve(\{sys,u(0)=u0,v(0)=v0,D(u)(0)=sol2,D(v)(0)=subs(b1=sol 2,b2)\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 " \+ \{u(t),v(t)\},type=numeric,output=listprocedure); " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 " u1:=subs(desys1,u(t)); v1:=subs( desys1,v(t));\n u2:=subs(desys2,u(t)); v2:=subs(desys2,v(t)); \+ GEO1[i]:=spacecurve(subs(\{u='u1'(t),v='v1'(t)\},f(u,v)),t=tstar t..tend,color=red,thickness=2): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 " GEO2[i]:=spacecurve(subs(\{u='u2'(t),v='v2'(t)\},f(u,v)),t=tstar t..tend,color=red,thickness=2): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " POINT:=pointplot3d(f(u0 ,v0),color=black,symbol=circle,symbolsize=15,thickness=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 184 " SURF:=plot3d(subs(\{u=u(t),v=v(t)\},f(u,v )),u=ustart..uend,v=vstart..vend,color=grey,style=wireframe):\n displa y(seq(GEO1[i],i=0..iquo(n,2)),seq(GEO2[i],i=0..iquo(n,2)),POINT,SURF); \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "geodesic_plot(hel icoid,Pi,1,0,1,0,2*Pi,-4,4,-5,3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "geodesic_plot(helicoid,Pi,1,1,1,0,2*Pi,-4,4,- 4,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "geodesics_plot(hel icoid,Pi,1,0,2*Pi,-4,4,0,3,15);" }}}}{MARK "10 5 0" 36 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }