{VERSION 5 0 "SUN SPARC SOLARIS" "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 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "Math 4530" }}{PARA 258 "" 0 "" {TEXT -1 29 "4.2.8 - HW solutions to set 7" }}{PARA 256 "" 0 "" {TEXT -1 42 "Computations related to the s hape operator" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 262 "Here is a list of procedures to calculate the matrix of \+ the shape operator, the principle curvatures, the mean curvature and t he Gauss curvature, using a given patch X. The procedures are illustra ted with computations and pictures for the helicoid and the torus." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:\nwith(linalg):\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "assume(u,real); #this gets rid of that annoying \"c sgn\" fcn\nassume(v,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "#dot product\ndp:=proc(X,Y)\nX[1]*Y[1]+X[2]*Y[2]+X[3]*Y[3];\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "#2-norm, i.e. magnitude .\nnrm:=proc(X)\nsqrt(dp(X,X));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "#cross product:\nxp:=proc(X,Y)\nlocal a,b,c;\na:=X[2 ]*Y[3]-X[3]*Y[2];\nb:=X[3]*Y[1]-X[1]*Y[3];\nc:=X[1]*Y[2]-X[2]*Y[1];\n[ a,b,c];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "#Derivati ve matrix for mapping X:\nDXq:=proc(X)\nlocal Xu,Xv;\nXu:=matrix(3,1,[ diff(X[1],u),diff(X[2],u),diff(X[3],u)]);\nXv:=matrix(3,1,[diff(X[1],v ),diff(X[2],v),diff(X[3],v)]);\nsimplify(augment(Xu,Xv),radical,symbol ic,trig);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "#Matrix of first fundamental form:\ngij:=proc(X)\nlocal g11,g12,g22,Y;\nY:=ev alm(DXq(X));\nsimplify(evalm(transpose(Y)&*Y),\n radical,symbolic,tr ig);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "#unit normal :\nU:=proc(X)\nlocal Y,Z,s;\nY:=DXq(X);\nZ:=xp(col(Y,1),col(Y,2));\ns: =nrm(Z);\nsimplify(evalm((1/s)*Z),radical,symbolic,trig);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 412 "#matrix of second fundament al form:\nhij:=proc(X)\nlocal Y,Xu,Xv,Xuu,Xuv,Xvv,U1,h11,h12,h22;\nY:= DXq(X);\nU1:=U(X);\nXu:=col(Y,1);\nXv:=col(Y,2);\nXuu:=[diff(Xu[1],u), diff(Xu[2],u),diff(Xu[3],u)];\nXuv:=[diff(Xu[1],v),diff(Xu[2],v),diff( Xu[3],v)];\nXvv:=[diff(Xv[1],v),diff(Xv[2],v),diff(Xv[3],v)];\nh11:=dp (Xuu,U1);\nh12:=dp(Xuv,U1);\nh22:=dp(Xvv,U1);\nsimplify(matrix(2,2,[h1 1,h12,h12,h22]),\n radical,symbolic,trig);\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "#matrix of shape operator wrt basis \{Xu ,Xv\}:\naij:=proc(X)\nlocal Y,H,G;\nH:=hij(X);\nG:=gij(X);\nsimplify(e valm(inverse(G)&*H),\n radical,symbolic,trig);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "#Gauss curvature\nGK:=proc(X)\nloca l A;\nA:=aij(X);\nsimplify(det(A),radical,symbolic,trig);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "#Mean curvature\nMK:=proc(X) \nlocal A;\nA:=aij(X);\nsimplify(1/2*trace(A),radical,symbolic,trig); \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "#Principle curvat ures and directions:\nPK:=proc(X)\nlocal Y;\nY:=aij(X);\neigenvects(Y) ;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "test:=[u,v,u^2-v ^2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "DXq(test);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "gij(test);\nhij(test);\nsub s(\{u=0,v=0\},aij(test));\nsubs(\{u=0,v=0\},GK(test));\nsubs(\{u=0,v=0 \},MK(test));\nsubs(\{u=0,v=0\},aij(test));\n" }}}{PARA 0 "" 0 "" {TEXT 256 18 "4.2.8 begins here:" }}{PARA 0 "" 0 "" {TEXT -1 11 "(a) H elcat:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assume(t,real);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "helcat:=[cos(t)*sinh(v)*s in(u)+sin(t)*cosh(v)*cos(u),\n-cos(t)*sinh(v)*cos(u)+sin(t)*cosh(v)*si n(u),\nu*cos(t)+v*sin(t)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "GK(helcat);\nMK(helcat);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "animate3d(helcat,u=0..2*Pi,v=-1..1,t=0..Pi/2,color=GK(helcat), \nscaling=constrained);#this lets you see the helicoid lowering\n#itse lf into a catenoid. By coloring with GK it is easier\n#to see corresp onding points." }}}{PARA 0 "" 0 "" {TEXT -1 49 "b) Henneberg. (Maple \+ could not compute MK or GK)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "henne:=[2*sinh(u)*cos(v)-2/3*sinh(3*u)*cos(3*v),\n2*sinh(u)*sin(v )+2/3*sinh(3*u)*sin(3*v),\n2*cosh(2*u)*cos(2*v)];" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 77 "plot3d(henne,u=-1..1,v=0..2*Pi,grid=[30,30], \nscaling=constrained,axes=boxed);" }}}{PARA 0 "" 0 "" {TEXT -1 54 "c) Catalan: Maple couldn't compute GK or MK for this!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "catalan:=[u-sin(u)*cosh(v),1-cos(u)*cosh( v),\n4*sin(u/2)*sinh(v/2)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot3d(catalan,u=0..4*Pi,v=-1.5..1.5,grid=[30,30],\nscaling=constr ained,axes=boxed);" }}}{PARA 0 "" 0 "" {TEXT -1 12 "d) Enneper:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "enne:=[u-u^3/3+u*v^2,v-v^3/3 +v*u^2,u^2-v^2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "GK(enne );\nMK(enne);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot3d(enn e,u=-1.5..1.5,v=-1.5..1.5,grid=[30,30],\nscaling=constrained,axes=boxe d);" }}}{PARA 0 "" 0 "" {TEXT -1 25 "e) Sherk's Fifth surface:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sherk:=[arcsinh(u),arcsinh(v ),arcsin(u*v)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "GK(sherk );\nMK(sherk);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plot3d(sh erk,u=-1..1,v=-1..1,grid=[30,30],\nscaling=constrained,axes=boxed);" } }}{PARA 0 "" 0 "" {TEXT -1 29 "f) Planar lines of curvature:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "planarlines:=[1/sqrt(1-t^2) *(t*u+sin(u)*cosh(v)),\n1/sqrt(1-t^2)*(v+t*cos(u)*sinh(v))\n,cos(u)*co sh(v)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "GK(planarlines); \nMK(planarlines);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "anima te3d(planarlines,u=0..2*Pi,\nv=-1..1,t=0..(.5),\nscaling=constrained); #at t=0 it's a catenoid!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "animate3d(planarlines,u=0..2*Pi,\nv=-1..1,t=0..(.5),\nscaling=con strained); #which starts\n#getting pulled as t increases!" }}}}{MARK " 48" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }