(1) Execute the module Surf() defined in the next section. Once executed the commands in the module are active. (2) To call one of the functions in the module ( EFG, UN, lmn, shape,shape_matrix,normal_curvature,GK,MK) use, for example, the notation Surf:-EFG(X)

Surf:= module() export EFG, UN, lmn, shape,shape_matrix,normal_curvature,GK,MK; # The metric for the surface EFG:= proc(X) local Xu,Xv,E,F,G; Xu:= <diff(X[1],u),diff(X[2],u),diff(X[3],u)>; Xv:= <diff(X[1],v),diff(X[2],v),diff(X[3],v)>; E:= LinearAlgebra[DotProduct](Xu,Xu,conjugate=false); F:= LinearAlgebra[DotProduct](Xu,Xv,conjugate=false); G:= LinearAlgebra[DotProduct](Xv,Xv,conjugate=false); simplify([E,F,G]); end proc: # The unit normal for the surface UN:= proc(X) local Xu,Xv,Z,s; Xu:= <diff(X[1],u),diff(X[2],u),diff(X[3],u)>; Xv:= <diff(X[1],v),diff(X[2],v),diff(X[3],v)>; Z:= LinearAlgebra[CrossProduct](Xu,Xv); s:= LinearAlgebra[VectorNorm](Z,Euclidean,conjugate=false); simplify(<Z[1]/s,Z[2]/s,Z[3]/s>,sqrt,trig,symbolic); end proc: #Coefficients of the second fundamental form lmn:= proc(X) local Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n; Xu:= <diff(X[1],u),diff(X[2],u),diff(X[3],u)>; Xv:= <diff(X[1],v),diff(X[2],v),diff(X[3],v)>; Xuu:= <diff(Xu[1],u),diff(Xu[2],u),diff(Xu[3],u)>; Xuv:= <diff(Xu[1],v),diff(Xu[2],v),diff(Xu[3],v)>; Xvv:= <diff(Xv[1],v),diff(Xv[2],v),diff(Xv[3],v)>; U:= UN(X); l:= LinearAlgebra[DotProduct](U,Xuu,conjugate=false); m:= LinearAlgebra[DotProduct](U,Xuv,conjugate=false); n:= LinearAlgebra[DotProduct](U,Xvv,conjugate=false); simplify([l,m,n],sqrt,trig,symbolic); end proc: #The Shape operator shape:= proc(X) local Y,Z,a,b,c,d; Y:= EFG(X); Z:= lmn(X); a:= simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); b:= simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); c:= simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); d:= simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); [S(x_u)=a*x_u+b*x_v,S(x_v) = c*x_u+d*x_v]; end proc: #The matrix of the shape operator shape_matrix:= proc(X) local Y,Z,a,b,c,d; Y:= EFG(X); Z:= lmn(X); a:= simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); b:= simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); c:= simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); d:= simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2)); Matrix([[a,c],[b,d]]); end proc: #The normal curvature. (u0,v0) is the point and 't' is the angle normal_curvature:= proc(X,u0,v0,t) local S,BB; S:= subs({u=u0,v=v0},shape_matrix(X)); BB:= LinearAlgebra[DotProduct](S.<cos(t),sin(t)>,<cos(t),sin(t)>,conjugate=false); simplify(BB,trig,symbolic); end proc; #Gaussian Curvatures GK:= proc(X) local E,F,G,l,m,n,S,T; S:= Surf:-EFG(X); T:= Surf:-lmn(X); E:= S[1]; F:= S[2]; G:= S[3]; l:= T[1]; m:= T[2]; n:= T[3]; simplify((l*n-m^2)/(E*G-F^2),sqrt,trig,symbolic); end; #Mean curvature MK:= proc(X) local E,F,G,l,m,n,S,T; S:= Surf:-EFG(X); T:= Surf:-lmn(X); E:= S[1];F:= S[2]; G:= S[3]; l:= T[1];m:= T[2]; n:= T[3]; simplify((G*l+E*n-2*F*m)/(2*E*G-2*F^2),sqrt,trig,symbolic); end; end module;

The shape operator on a basis, and shape matrix for the sphere: TheSphere:= <R*cos(u)*cos(v)|R*sin(u)*cos(v)|R*sin(v)>; Surf:-shape(<R*cos(u)*cos(v)|R*sin(u)*cos(v)|R*sin(v)>); Surf:-shape_matrix(<R*cos(u)*cos(v)|R*sin(u)*cos(v)|R*sin(v)>); Surf:-normal_curvature(<R*cos(u)*cos(v)|R*sin(u)*cos(v)|R*sin(v)>,0,0,t); Surf:-MK(TheSphere); Surf:-GK(TheSphere); Here is the saddle surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNictRiw2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYoRistRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGNkY5LUkjbW5HRiQ2JFEiMkYnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZlby1GIzYkLUZlbjYlLUYsNiVRInlGJ0Y2RjlGam5GXm9GQEYrRkBGK0ZARitGQEYrRkA=. XX:= <u,v,u^2-v^2>; Surf:-shape(XX); Surf:-shape_matrix(XX); Surf:-normal_curvature(XX,0,0,Pi/2); Surf:-normal_curvature(XX,0,0,0); Surf:-GK(XX); Surf:-MK(XX); Here is the complicated surfaced z = x*y. TheSurface:= <u,v,u*v>; Surf:-shape(<u,v,u*v>); A:= Surf:-shape_matrix(<u,v,u*v>); LinearAlgebra[Eigenvalues](A); LinearAlgebra[Determinant](A); Surf:-GK(TheSurface); Surf:-MK(TheSurface);

XY:= <u,v,u^4+v^4-4*u*v+1>;

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Surf:-normal_curvature(XY,0,0,3*Pi/4);

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TTdSMApJNFJUQUJMRV9TQVZFLzQzOTEwMjRYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQlInVHJSJ2RywqKiRGJyIiJSIiIiokRihGK0YsKiZGJ0YsRihGLCEiJUYsRixGJg==