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:

sph:=<R*cos(u)*cos(v)|R*sin(u)*cos(v)|R*sin(v)>;

EFG(sph);

lmn(sph);

GK(sph);

MK(sph);

tor1:=<(R+r*cos(u))*cos(v)|(R+r*cos(u))*sin(v)|r*sin(u)>;

GK(tor1);

MK(tor1);

catenoid:=<cosh(u)*cos(v)|cosh(u)*sin(v)|u>;

GK(catenoid);MK(catenoid);

ellpar:=<u|v|u^2/a^2+v^2/b^2>;

GK(ellpar);MK(ellpar);

helicoid:=<u*cos(v)|u*sin(v)|v>;

plot3d(helicoid,u=0..1.5,v=0..5*Pi,color=GK(helicoid),
> grid=[10,40]);

pair:=<u|v|ln((u^4+2*u^2*v^2-2*u^2+v^4+2*v^2+1)^(1/2))>;

GK(pair);

MK(pair);

plot3d(pair,u=-1.5..1.5,v=-1.5..1.5,axes=boxed,shading=xy,
> lightmodel=light2,grid=[30,30],orientation=[-33,61]);

plot3d(GK(pair),u=-1.5..1.5,v=-1.5..1.5,axes=boxed);

plot3d(MK(pair),u=-1.5..1.5,v=-1.5..1.5,axes=boxed);

plot3d(pair,u=-1.5..1.5,v=-1.5..1.5,scaling=constrained,
> color=GK(pair),orientation=[-33,61],grid=[30,30]);

plot3d(pair,u=-1.5..1.5,v=-1.5..1.5,scaling=constrained,
> color=MK(pair),orientation=[83,147],grid=[30,30]);

plotposconstcurv:=proc(c,theta,phi)
> local X,desys,gu,s;
> if 1/c>1 then s:=-Pi/2..Pi/2 else s:=-arcsin(1/c)..
> arcsin(1/c) fi;
> X:=[c*cos(u)*cos(v),c*cos(u)*sin(v),g];
> desys:=dsolve({diff(g(u),u)=sqrt(1-c^2*sin(u)^2),g(0)=0},
> g(u),type=numeric,output=listprocedure):
> gu:=subs(desys,g(u));
> plot3d(subs(g='gu'(u),X),u=s,v=0..2*Pi,scaling=constrained,
> style=patch,shading=zhue,orientation=[theta,phi]);
> end:

plotposconstcurv(1,49,78);

plotposconstcurv(0.65,49,78);

plotposconstcurv(1.7,49,69);

plotnegconstcurv:=proc(c,d,theta,phi)
> local X,desys,gu,s,h,kk,init;
> if d=0 then h:=c*exp(u);
> s:=-2.5..ln(1/c);
> init:=-1 fi;
> if c>0 and d>0 then h:=2*sqrt(c*d)*cosh(u);
> s:=arcsinh(-1/(2*sqrt(c*d)))..arcsinh(1/(2*sqrt(c*d)));
> init:=0 fi;
> if c>=0 and d<0 then h:=2*sqrt(-c*d)*cosh(u);
> kk=c*d;
> if kk>-1/4 then s:=0..arccosh(1/(2*sqrt(-c*d)))fi;
> init:=0 fi;
> X:=[h*cos(v),h*sin(v),g];
> desys:=dsolve({diff(g(u),u)=sqrt(1-diff(h,u)^2),g(init)=0},
> g(u),type=numeric,output=listprocedure):
> gu:=subs(desys,g(u));
> plot3d(subs(g='gu'(u),X),u=s,v=0..2*Pi,scaling=constrained,
> style=patch,shading=zhue,grid=[15,40],orientation=
> [theta,phi]);
> end:

plotnegconstcurv(1,0,41,64);

plotnegconstcurv(0.3,1,47,64);

plotnegconstcurv(0.6,0.1,49,69);

with(plots):with(LinearAlgebra):

h:=t->h(t);g:=t->g(t);