{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 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 "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 0 "" 0 "" {TEXT -1 348 "This mapleworksheet plots surfaces according to their curvatures. So if nearby points have dif ferent curvatures, they will be plotted with different colors. Of cou rse, surfaces can be plotted using color schemes derived from Gaussian Curvature and from Mean Curvature. Interesting properties about surf aces are revealed by these curvature plots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 674 "with(plots):\nwith(LinearAlgebra):\nGK:=proc(x)\n local xu,xv,E,F,G,e,f,g,xuu,xvv,xuv,N,k,mk;\nxu:=;\nxv:=;\nE :=DotProduct(xu,xu,conjugate=false);\nF:=DotProduct(xu,xv,conjugate=fa lse);\nG:=DotProduct(xv,xv,conjugate=false);\nxuu:=;\nxuv:=;\nxvv:=;\nN:=CrossPro duct(xu,xv)/sqrt(E*G-F^2);\ne:=DotProduct(N,xuu,conjugate=false);\nf:= DotProduct(N,xuv,conjugate=false);\ng:=DotProduct(N,xvv,conjugate=fals e);\nk:=(e*g-f^2)/(E*G-F^2);\nmk:=(G*e+E*g-2*F*f)/(2*E*G-2*F^2);\nsimp lify(k);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 662 "with(Lin earAlgebra):\nMK:=proc(x)\nlocal xu,xv,E,F,G,e,f,g,xuu,xvv,xuv,N,k,mk; \nxu:=;\nxv:=;\nE:=DotProduct(xu,xu,conjugate=false);\nF:=Do tProduct(xu,xv,conjugate=false);\nG:=DotProduct(xv,xv,conjugate=false) ;\nxuu:=;\nxuv:=;\nxvv:=;\nN:=CrossProduct(xu,xv)/sqrt(E*G-F^2);\ne:=DotProduct(N ,xuu,conjugate=false);\nf:=DotProduct(N,xuv,conjugate=false);\ng:=DotP roduct(N,xvv,conjugate=false);\nk:=(e*g-f^2)/(E*G-F^2);\nmk:=(G*e+E*g- 2*F*f)/(2*E*G-2*F^2);\nsimplify(mk);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "First, we plot surfaces according to their Gaussian Curva tures." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "helicoid:=[u*cos( v),u*sin(v),v];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)helicoidG7%*&%\" uG\"\"\"-%$cosG6#%\"vGF(*&F'F(-%$sinGF+F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot3d(helicoid,u=0..1.5,v=0..5*Pi,color=GK(helico id));" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6#-%%MESH G6$X.%)anythingG6\"6\"[gl'!%\"!!$^`r\":\":\"$0000000000000000000000000 00000000000000000000000000000000000000000000000000000003FE4F1A6C638D04 B000000000000000000000000000000003FF4F1A6C638D04B800000000000000000000 000000000003FFF6A7A29553870800000000000000000000000000000004004F1A6C63 8D04B80000000000000008000000000000000400A2E1077C7045E80000000000000008 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