with(plots):with(LinearAlgebra):

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:=DotProduct(Xu,Xu,conjugate=false);
> F:=DotProduct(Xu,Xv,conjugate=false);
> G:=DotProduct(Xv,Xv,conjugate=false);
> simplify([E,F,G]);
> end:

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:=CrossProduct(Xu,Xv);
> s:=VectorNorm(Z,Euclidean,conjugate=false);
> simplify(<Z[1]/s|Z[2]/s|Z[3]/s>,sqrt,trig,symbolic);
> end:

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:=DotProduct(U,Xuu,conjugate=false);
> m:=DotProduct(U,Xuv,conjugate=false);
> n:=DotProduct(U,Xvv,conjugate=false);
> simplify([l,m,n],sqrt,trig,symbolic);
> end:

GK:=proc(X)
> local E,F,G,l,m,n,S,T;
> S:=EFG(X);T:=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:

MK:=proc(X)
> local E,F,G,l,m,n,S,T;
> S:=EFG(X);T:=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:

helcat:=<cos(t)*sinh(v)*sin(u)+sin(t)*cosh(v)*cos(u)|-cos(t)*sinh(v)*cos(u)+sin(t)*cosh(v)*sin(u)
> |u*cos(t)+v*sin(t)>;

MK(helcat);

GK(helcat);

henneberg:=<2*sinh(u)*cos(v)-2/3*sinh(3*u)*cos(3*v)|2*sinh(u)*sin(v)+2/3*sinh(3*u)*sin(3*v)|2*cosh(2*u)*cos(2*v)>;

MK(henneberg);

GK(henneberg);

catalan:=<u-sin(u)*cosh(v)|1-cos(u)*cosh(v)|4*sin(u/2)*sinh(v/2)>;

MK(catalan);

GK(catalan);

Ennep:=<u-u^3/3+u*v^2|v-v^3/3+v*u^2|u^2-v^2>;

MK(Ennep);

GK(Ennep);

sherk5:=<arcsinh(u)|arcsinh(v)|arcsin(u*v)>;

MK(sherk5);

GK(sherk5);

plane:=<(c*u+sin(u)*cosh(v))/sqrt(1-c^2)|(v+c*cos(u)*sinh(v))/sqrt(1-c^2)|cos(u)*cosh(v)>;

MK(plane);

GK(plane);