{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 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 311 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 316 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 317 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 321 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 322 "" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 325 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 326 "" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 327 "" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 1 10 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 0 10 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 330 "" 1 10 0 0 0 0 0 1 0 0 0 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 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 323 0 "" }{TEXT 324 23 "Calculus IV with Maple\n" }{TEXT 325 32 "Copyright 2002, Dr. J ack Wagner\n" }{TEXT -1 30 "j.wagner@intelligentsearch.com" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 19 "Lesson 6: Curvature" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 9 "Topics: " }{TEXT 322 63 "Normal curvature, Gaussian curvature, second fundamental form, " }{TEXT 326 16 "Dupin Indicatrix" }{TEXT 327 3 ", \n" }{TEXT 329 25 "Maple commands introduced" }{TEXT 330 2 ": " }{TEXT 328 40 "Principal Normal, BilinearForm, Curvature" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 256 16 "Normal Curvature" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Consider a surf ace, " }{XPPEDIT 18 0 "F(u,v) = [u, v, u^2-v^2];" "6#/-%\"FG6$%\"uG%\" vG7%F'F(,&*$F'\"\"#\"\"\"*$F(F,!\"\"" }{TEXT -1 42 " in which we embed the image of the curve " }{XPPEDIT 18 0 "h(t) = [t, cos(t)];" "6#/-% \"hG6#%\"tG7$F'-%$cosG6#F'" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart: with(VectorCalculus): with(LinearAlgebra) :with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot3d(F, u= -2..2, v=-2..2, axes=normal, color=aquamarine, style=wireframe);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P := ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "C := eval(F, \{u=t, v=cos(t) \});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "P0 := spacecurve(ev alm(P), t=-2..2, color=blue, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 82 "P1 := plot3d(F, u=-2..2, v=-2..2, axes=normal, colo r=aquamarine, style=wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "P2 := spacecurve(evalm(C), t=-2..2, color=red, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(P0, P1, P2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Now define a map, " }{XPPEDIT 18 0 "N(u,v) = F[u]*x*F[v]/abs(F[u]*x*F[v]);" "6#/-%\"NG6$%\"uG%\"vG**&% \"FG6#F'\"\"\"%\"xGF-&F+6#F(F--%$absG6#*(&F+6#F'F-F.F-&F+6#F(F-!\"\"" }{TEXT -1 148 " . This is the unit normal to the surface and is a map \+ to the unit sphere. Restricted to C it is a curve embedded in the surf ace of the unit sphere." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dF[u] := diff(F, u);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dF[v] := diff(F, v);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nn := (dF[u] &x dF[v]);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N := Normalize(nn, 2, inpla ce):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N := simplify(N) as suming real;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Nt := eval( N, \{u=t, v=cos(t)\});" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The normal vector N(u,v) shown as a parametrized surface over u and v ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "P3 := plot3d(N, u=-2. .2, v=-2..2, color=aquamarine, axes=normal, style=wireframe, scaling=c onstrained, numpoints=2000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "P4 := spacecurve(evalm(Nt), t=-2..2, axes=normal, color=red, thi ckness=2, style=wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "display(P3, P4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dC \+ := diff(C, t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dC0 := ev al(dC, t=0); #The tangent to C at t=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "C0 := eval(C, t=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "P5 := spacecurve(evalm(C0+s*dC0), s=-1..1, color=blue , thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display( P1, P2, P5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "The differential of the map N, dN, is a map from the tangent space of the F(u, v) surf ace to the tangent space of the unit sphere. Since N.N=1 -> N'.N = 0, dN lives in the tangent space. dN = " }{XPPEDIT 18 0 "N[u]*du;" "6#*& &%\"NG6#%\"uG\"\"\"%#duGF(" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "N[v]*dv; " "6#*&&%\"NG6#%\"vG\"\"\"%#dvGF(" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "JN := Jacobian(N, [u, v, w]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Let dNt \+ be the map dN restricted to the curve C(t) = F(P(t)). dNt = N(C(t))' = N'(C(t))C'(t). This is the tangent to the image of the curve C(t) on \+ the unit sphere. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "JNt := simplify( eval(JN, \{u=t, v=cos(t)\}) );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "dP := diff(P, t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "JNt0 := eval(JNt, t=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Tt := simplify(JNt.dP); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Evaluating at t=0; The im age of the tangent to C at t=0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " Tt0 := eval(Tt, t=0); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "At the point N(0, 1) corresponding \+ to t=0, the image of dC0 under the map dNt is the tangent to the imag e of C under the map N." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " N0 := eval(Nt, t=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "P6 \+ := spacecurve(evalm(N0+s*Tt0), s=-1/2..1/2, color=blue, thickness=2): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(P3, P4, P6);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Now consider the principal norma l vector to the embedded curve C." }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "PrincipalNormal" 2 "PrincipalNormal" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "n := simplify( Princip alNormal(C, t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "n0 := \+ eval(n, t=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Note that the principal normal, n, is " }{TEXT 257 3 " not" }{TEXT -1 21 " a unit vector. n = " }{XPPEDIT 18 0 "kappa;" "6#% &kappaG" }{TEXT -1 1 " " }{TEXT 258 4 "n , " }{TEXT -1 6 "where " } {TEXT 259 1 "n" }{TEXT -1 20 " is the unit normal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "P7 := spacecurve(\{evalm(C0+s*Tt0), evalm( C0+3*s*N0), evalm(C0+s*n0)\}, s=-1..1, color=blue, thickness=3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(P1, P2, P7);" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "D efine the " }{TEXT 260 17 "normal curvature " }{TEXT -1 31 "as the pro jection of n onto N. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ka ppa['n'] := N0.n0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Because N i s a unit vector, = = " }{XPPEDIT 18 0 "kapp a;" "6#%&kappaG" }{TEXT -1 8 " cos(N, " }{TEXT 262 1 "n" }{TEXT -1 9 " ), where " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1 26 " is the \+ usual curvature. " }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Curvature " 2 "Curvature" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "k := eval(Curvature(C, t), t=0);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "cos(theta) := N0.Normalize(n0, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(k*cos(theta));" }}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 23 " Second Fundamental Form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We now define a bilinear form " } {XPPEDIT 18 0 "II[p](tau);" "6#-&%#IIG6#%\"pG6#%$tauG" }{TEXT -1 20 ". Set dN = JNt. Let " }{TEXT 274 1 "x" }{TEXT -1 7 "(s) = " }{XPPEDIT 18 0 "[u(s), v(s)];" "6#7$-%\"uG6#%\"sG-%\"vG6#F'" }{TEXT -1 68 " be a curve in the surface F(u, v), parametrized by arc length. Let " } {XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 40 "(s) be the unit tangent at s. Define: " }{XPPEDIT 18 0 "II[p];" "6#&%#IIG6#%\"pG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 6 "(s)) =" }{TEXT 271 1 " " }{TEXT 278 0 "" }{TEXT -1 1 "-" }{TEXT 277 1 " " }{TEXT 272 3 " " }{XPPEDIT 18 0 "II[p];" "6#&%#IIG6#%\"pG" } {TEXT -1 1 "(" }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 19 "(s)) is \+ called the " }{TEXT 275 24 "second fundamental form " }{TEXT -1 37 "of F at p. We will work at the point " }{TEXT 307 1 "p" }{TEXT -1 3 " = \+ " }{TEXT 276 1 "x" }{TEXT -1 11 "(0). Let N(" }{TEXT 308 1 "x" }{TEXT -1 32 "(s)) be the restriction of N to " }{TEXT 309 1 "x" }{TEXT -1 5 "(s). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " Since N(" }{TEXT 310 1 "x" }{TEXT -1 31 "(0)) is normal to the tangent " }{TEXT 311 2 "x'" }{TEXT -1 15 "(0) we have =0. " }}{PARA 0 "" 0 "" {TEXT -1 4 " + = 0 -> = - = " }{XPPEDIT 18 0 "kappa[n];" "6#&%&kappaG6#%\"nG" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Implementing this in Maple:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "II[p] := JNt0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u := t: v := cos(t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "du := diff(u, t); dv := diff(v, t); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dx := ;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Note that at t=0 this is a " } {TEXT 296 4 "unit" }{TEXT -1 19 " tangent vector. " }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "BilinearForm" 2 "BilinearForm" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B := -BilinearForm (dx, dx, JNt0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Bt0 := e val(B, t=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Expand " } {XPPEDIT 18 0 "II[p];" "6#&%#IIG6#%\"pG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "tau(s);" "6#-%$tauG6#%\"sG" }{TEXT -1 3 ") =" }{TEXT 263 1 " " } {TEXT 280 0 "" }{TEXT -1 1 "-" }{TEXT 279 1 " " }{TEXT 268 3 "" }{TEXT 267 2 "= " }{TEXT 282 0 "" }{TEXT -1 1 "-" } {TEXT 281 1 " " }{TEXT 269 0 "" }{TEXT -1 0 "" }{TEXT 265 0 "" }{TEXT -1 1 "(" }{TEXT 264 1 " " }{TEXT -1 0 "" }{XPPEDIT 18 0 "N[u]*du+N[v]* dv;" "6#,&*&&%\"NG6#%\"uG\"\"\"%#duGF)F)*&&F&6#%\"vGF)%#dvGF)F)" } {TEXT -1 4 ").( " }{XPPEDIT 18 0 "x[u]*du;" "6#*&&%\"xG6#%\"uG\"\"\"%# duGF(" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "x[v]*dv;" "6#*&&%\"xG6#%\"vG \"\"\"%#dvGF(" }{TEXT -1 6 ") = - " }{XPPEDIT 18 0 "`<,>`(N[u],x[u])*d u^2;" "6#*&-%$<,>G6$&%\"NG6#%\"uG&%\"xG6#F*\"\"\"*$%#duG\"\"#F." } {TEXT -1 5 " - " }{XPPEDIT 18 0 "`<,>`(N[v],x[u])*du*dv;" "6#*(-%$<, >G6$&%\"NG6#%\"vG&%\"xG6#%\"uG\"\"\"%#duGF/%#dvGF/" }{TEXT -1 4 " - \+ " }{XPPEDIT 18 0 "`<,>`(N[u],x[v])*du*dv;" "6#*(-%$<,>G6$&%\"NG6#%\"uG &%\"xG6#%\"vG\"\"\"%#duGF/%#dvGF/" }{TEXT -1 5 " - " }{XPPEDIT 18 0 "`<,>`(N[v],x[v])*dv^2;" "6#*&-%$<,>G6$&%\"NG6#%\"vG&%\"xG6#F*\"\"\"*$ %#dvG\"\"#F." }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{TEXT 266 1 "N" }{TEXT -1 38 " is orthogonal to the tangent plane, " }{XPPEDIT 18 0 "`<,>`(N,x[u]);" " 6#-%$<,>G6$%\"NG&%\"xG6#%\"uG" }{TEXT -1 8 " = 0 -> " }{XPPEDIT 18 0 " `<,>`(N[u],x[u]);" "6#-%$<,>G6$&%\"NG6#%\"uG&%\"xG6#F)" }{TEXT -1 4 " \+ = -" }{XPPEDIT 18 0 "`<,>`(N,x[uu]);" "6#-%$<,>G6$%\"NG&%\"xG6#%#uuG" }{TEXT -1 3 " ->" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "II[p];" "6#&%#IIG6# %\"pG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "tau(s);" "6#-%$tauG6#%\"sG" } {TEXT -1 4 ") = " }{XPPEDIT 18 0 "`<,>`(N,x[uu])*du^2;" "6#*&-%$<,>G6$ %\"NG&%\"xG6#%#uuG\"\"\"*$%#duG\"\"#F," }{TEXT -1 4 " +2 " }{XPPEDIT 18 0 "`<,>`(N,x[uv])*du*dv;" "6#*(-%$<,>G6$%\"NG&%\"xG6#%#uvG\"\"\"%#d uGF,%#dvGF," }{TEXT -1 3 " + " }{XPPEDIT 18 0 "`<,>`(N,x[vv])*dv^2;" " 6#*&-%$<,>G6$%\"NG&%\"xG6#%#vvG\"\"\"*$%#dvG\"\"#F," }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Set L= " }{XPPEDIT 18 0 "`<,>`(N,x[uu])" " 6#-%$<,>G6$%\"NG&%\"xG6#%#uuG" }{TEXT -1 7 " , M = " }{XPPEDIT 18 0 "` <,>`(N,x[uv])" "6#-%$<,>G6$%\"NG&%\"xG6#%#uvG" }{TEXT -1 7 " , N = " } {XPPEDIT 18 0 "`<,>`(N,x[vv])" "6#-%$<,>G6$%\"NG&%\"xG6#%#vvG" }{TEXT -1 55 " The second fundamental form may then be expressed as:" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "II[p];" "6#&%#IIG6#%\"pG" }{TEXT -1 1 " (" }{XPPEDIT 18 0 "tau(s);" "6#-%$tauG6#%\"sG" }{TEXT -1 6 ") = L " } {XPPEDIT 18 0 "du^2;" "6#*$%#duG\"\"#" }{TEXT -1 6 " + 2M " }{XPPEDIT 18 0 "du*dv;" "6#*&%#duG\"\"\"%#dvGF%" }{TEXT -1 5 " + N " }{XPPEDIT 18 0 "dv^2;" "6#*$%#dvG\"\"#" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart: with(VectorCalculus): with(LinearAlgebr a):with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dF[u] := d iff(F, u); dF[v] := diff(F, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n := dF[u] &x dF[v];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "NN := Normalize(n, 2, inplace):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "NN := simplify(NN) assuming real;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dF[uu] := diff(dF[u], u);" }}}{EXCHG } {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dF[uv] := diff(dF[u], v);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dF[vv] := diff(dF[v], v); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L := NN.dF[uu];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "M := NN.dF[uv];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "N := NN.dF[vv];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u := t: v := cos(t):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "du := diff(u, t); dv := diff(v, t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "II[p] := L*(du)^2+M*du*dv+N* (dv)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eval(II[p], t=0) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart:with(plots): w ith(VectorCalculus): with(LinearAlgebra):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The maximum and minimum c urvatures are the eigenvalues of " }{TEXT 283 3 "dN." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := ;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dF[u] := diff(F, u); dF[v] \+ := diff(F, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n := dF[u ] &x dF[v];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "N := Normali ze(n, 2, inplace):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N := \+ simplify(N) assuming real;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dN := Jacobian(N, [u, v, w]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dNt := subs(\{u=t, v=cos(t)\}, dN):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dNt0 := subs(t=0, dNt);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E := Eigenvalues(dNt0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "kappa[1][t=0] := E[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "kappa[2][t=0] := E[3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 " The eigenvectors of " }{TEXT 284 2 "dN" } {TEXT -1 9 " are the " }{TEXT 285 21 "principal directions." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E := Ei genvectors(dNt0, output='vectors');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e[1] := <0, -1/2, 1>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "e[2] := <1, 0, 0>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "C := subs(\{u=t, v=cos(t)\}, F);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "C0 := subs(t=0, C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "P1 := plot3d(F, u=-2..2, v=-2..2, color=aquamari ne, style=wireframe, axes=normal):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "P2 := spacecurve(\{evalm(C0+s*e[1]), evalm(C0+s*e[2]) \}, s=-1..1, color=red, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "P3 := spacecurve(evalm(C), t=-2..2, color=blue, thick ness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(P1, P2, P3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 4 "" 0 "" {TEXT 286 18 "Gaussian Curvature" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: w ith(LinearAlgebra): with(VectorCalculus):with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The Gaussian curvature " }{TEXT 287 1 "K " }{TEXT -1 3 " = " }{XPPEDIT 18 0 "kappa[1]*kappa[2];" "6#*&&%&kappaG 6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 23 " . The mean curvature " }{TEXT 294 4 "H = " }{XPPEDIT 18 0 "(kappa[1]+kappa[2])/2;" "6#*&,&&%&kappaG6 #\"\"\"F(&F&6#\"\"#F(F(F+!\"\"" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 31 "In the coordinate basis of the " }{TEXT 290 13 "am bient space" }{TEXT -1 17 ", [u, v, w], let " }{XPPEDIT 18 0 "N = [n[1 ], n[2], n[3]];" "6#/%\"NG7%&%\"nG6#\"\"\"&F'6#\"\"#&F'6#\"\"$" } {TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "N := [n(u , v)[1], n(u, v)[2], n(u, v)[3]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dN := Jacobian(N, [u, v, w]);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "E := eigenvalues(dN);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "K := expand(E[2]*E[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "H := (E[2]+E[3])/2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Both " }{TEXT 291 1 "K" }{TEXT -1 5 " and " }{TEXT 292 1 " H" }{TEXT -1 116 " are independent of the z component of N, and are eq ual, respectively, to the determinant and trace of the minor of " } {XPPEDIT 18 0 "A[33];" "6#&%\"AG6#\"#L" }{TEXT -1 1 " " }{MPLTEXT 1 0 1 " " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "In the " } {TEXT 293 5 "local" }{TEXT -1 19 " coordinate basis, " }{XPPEDIT 18 0 "x[u],x[v];" "6$&%\"xG6#%\"uG&F$6#%\"vG" }{TEXT -1 2 " :" }}{PARA 0 " " 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "N[u] = a[11]*x[u]+a[12]*x[ v];" "6#/&%\"NG6#%\"uG,&*&&%\"aG6#\"#6\"\"\"&%\"xG6#F'F.F.*&&F+6#\"#7F .&F06#%\"vGF.F." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " \+ " }{XPPEDIT 18 0 "N[v] = a[21]*x[u]+a[22]*x[v];" "6#/&%\"NG6#%\"vG,&*& &%\"aG6#\"#@\"\"\"&%\"xG6#%\"uGF.F.*&&F+6#\"#AF.&F06#F'F.F." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A := Matrix([[a[1 1], a[12]], [a[21], a[22]]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "dN := A*Matrix([[x[u]], [x[v]]])=A.Matrix([[x[u]], [x[v]]]);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "I n this basis " }{TEXT 295 2 "dN" }{TEXT -1 27 " is given by the A matr ix. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The eigenvalues of the A \+ matrix are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e := Eigenva lues(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "K := expand(e[1 ]*e[2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 2 "K " }{TEXT -1 35 "is \+ the determinant of the A matrix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "H := (e[1]+e[2])/2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 2 "H " }{TEXT -1 13 "is 1/2 Tr(A)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{TEXT 298 1 "N" } {TEXT -1 28 " is a map from the surface, " }{TEXT 299 1 "F" }{TEXT -1 1 "(" }{TEXT 300 4 "u, v" }{TEXT -1 23 "), to the unit sphere, " } {TEXT 301 2 "dN" }{TEXT -1 37 " is a map from the tangent space of " }{TEXT 302 1 "F" }{TEXT -1 1 "(" }{TEXT 303 4 "u, v" }{TEXT -1 61 ") t o the tangent space of the sphere. More specifically, let " }{TEXT 305 1 "x" }{TEXT -1 55 " = [u(t), v(t)] be a curve embedded in the sur face and " }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 36 " be a unit t angent vector at point " }{TEXT 306 1 "p" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "I[p];" "6#&%\"IG6#%\"pG" }{TEXT -1 1 "(" } {XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 5 ") = <" }{XPPEDIT 18 0 "tau " "6#%$tauG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 8 "> " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "II[p];" "6#&%#IIG6#% \"pG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 6 ") \+ = -<" }{TEXT 304 3 " dN" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "tau" "6#%$tau G" }{TEXT -1 3 "), " }{XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 2 " >" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "II[p] = -dN*I[p];" "6#/&%#IIG6#%\"pG, $*&%#dNG\"\"\"&%\"IG6#F'F+!\"\"" }{TEXT -1 10 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "In matrix form, re collecting the designation of the coefficients of " }{XPPEDIT 18 0 "I[ p];" "6#&%\"IG6#%\"pG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "II[p];" "6# &%#IIG6#%\"pG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "N := 'N':F := 'F':E := 'E': " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "II_[p] := <, >;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "I_[p] := <, >;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A := A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Eq1 := II_[p]=-A*I_[p];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "invI_[p] := MatrixInverse(I_[p]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Eq2 := A=simplify(-II_[p] .invI_[p]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "This provides an e xplicit formula for the A matrix components. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 297 16 "Dupin Indicatrix" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 " For unit vectors, v, in the basis formed by the orthogonal (but not or thonormal) curvature directions, the locus of points:" }}{PARA 0 "" 0 "" {TEXT -1 7 "+/-1 = " }{XPPEDIT 18 0 "II[p](v);" "6#-&%#IIG6#%\"pG6# %\"vG" }{TEXT -1 35 " is known as the Dupin indicatrix. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart:with(plots): with(VectorCal culus): with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "assume(u, real): assume(v, real):\ninterface(showassumed=0):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := ;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dF[u] := diff(F, u); dF[v] : = diff(F, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n := dF[u] &x dF[v];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "N := Normaliz e(n, 2, inplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "N := s implify(N) assuming real;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dN := Jacobian(N, [u, v, w]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dNt := subs(\{u=t, v=cos(t)\}, dN):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dNt0 := subs(t=0, dNt);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E := Eigenvectors(dNt0, output='vectors') ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "e[1] := convert(Column (E, 1), Vector[row]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "e[ 2] := convert(Column(E, 2), Vector[row]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "V := VectorScalarMultiply(e[1], u)+VectorScalarMult iply(e[2], v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eq1 := 1= eval(BilinearForm(V, V, dNt0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eq2 := -1=eval(BilinearForm(V, V, dNt0));" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 142 "Actually, the Dupin indicatrix is defined on norm alized vectors but this would only complicate the arithmetic and add n othing to the picture. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " implicitplot(\{eq1, eq2\}, u=-5..5, v=-5..5, numpoints=1000);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "This gives substance to the desig nation \"hyperbolic point\" for those points on a surface with negati ve Gausian curvature." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Eg := Eigenvalues(dNt0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "K := Eg[2]*Eg[3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "In the case o f a point with positive Gaussian curvature:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with(LinearAlgebra): with(VectorCalculus) : with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "assume(u, real): assume(v, real):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G := ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dG[u] := diff(G, u); dG[v] := diff(G, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n := dG[u] &x dG[v];" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "N := Normalize(n, 2, inplace):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "N := simplify(N) assuming real;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dN := Jacobian(N, [u, v, w]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dNt := subs(\{u=t, v=cos(t) \}, dN):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dNt0 := subs(t= 0, dNt);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E := Eigenvecto rs(dNt0, output='vectors');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e[2] := <0, 1/2, 1>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "e[1] := <1, 0, 0>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "V := VectorScalarMultiply(e[1], u)+VectorScalarMultiply(e[2], v);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eq1 := 1=eval(BilinearForm(V , V, dNt0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eq2 := -1=e val(BilinearForm(V, V, dNt0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "implicitplot(\{eq1, eq2\}, u=-5..5, v=-5..5, numpoints=2000, s caling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "The desi gnation \"elliptic point\" for those points on a surface with positiv e Gaussian curvature seems appropriate." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Eg := Eigenvalues(dNt0);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "K := Eg[2]*Eg[3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Now we visualize the surface G(u, v), together with the Dupin \+ indicatrix and the curvature directions at the point u=0, v=1." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "First we recollect the curvature d irections." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e[1] := e[1]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e[2] := e[2];" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 " Construct a third orthogonal axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e[3] := e[2] &x e [1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 " The matrix of basis \+ transformation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "with(lin alg): T := stackmatrix(e[1], e[2], e[3]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "whattype(T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "T := convert(T, Matrix);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "The relationship between our local coordinate system and the ambie nt system is: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq := << e_[1], e_[2], e_[3]>> = T*<>;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The matrix of coordinate transformation: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "TTinv := Transpose(MatrixInv erse(T));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Solve the equation o f the indicatrix for one variable. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "V := solve(eq2, v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Define the two halves of the indicatrix as vectors." }{MPLTEXT 1 0 1 " " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " C1 := ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "C2 : = ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Transform the \+ vector components with the proper transformation matrix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ct1 := TTinv.C1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ct2 := TTinv.C2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Identify the point on the surface corresponding to u=0 , v=1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "G0 := subs(\{u=0, v=1\}, G);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Plot the indicatr ix and the coordinate directions at the indicated pointon the surface, together with a plot of the surface." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "P1 := spacecurve(\{evalm(G0+Ct1), evalm(G0+Ct2)\}, u =-5..5, numpoints=1000, color=blue, axes=normal, scaling=constrained): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "P2 := spacecurve(\{eval m(G0+s*e[1]), evalm(G0+s*e[2])\}, s=-2..2, color=red, thickness=2):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "P3 := plot3d(G, u=-1..1, v =0..1.5, color=aquamarine, style=wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(P1, P2, P3);" }}}}{MARK "0 0 2" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }