with(LinearAlgebra):with(plots):

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

plot3d(Ennep,u=-2..2,v=-2..2,scaling= constrained,shading=zhue,lightmodel=light1,orientation=[89,54]);

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:

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:

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:'

shape(<R*cos(u)*cos(v)|R*sin(u)*cos(v)|R*sin(v)>);

shape(<(R+r*cos(u))*cos(v),(R+r*cos(u))*sin(v),
> r*sin(u)>);

shape(Ennep);

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

shape(cylinder);

plot3d(subs(R=2,cylinder),u=0..2*Pi,v=0..3,shading=XY,
> lightmodel=light1,grid=[25,10],orientation=[-46,61]);

shape(<u,v,u*v>);

Eigenvalues(shape_matrix(<u,v,u*v>));

Determinant(shape_matrix(<u,v,u*v>));

shape(<cosh(u)*cos(v)|cosh(u)*sin(v)|u>);

surfrev:=<g(u)|h(u)*cos(v)|h(u)*sin(v)>;

shape_matrix(surfrev);

Determinant(shape_matrix(surfrev));

normal_curvature:=proc(X,u0,v0,t)
> local S;
> S:=subs({u=u0,v=v0},shape_matrix(X));
> DotProduct(S.<cos(t),sin(t)>,<cos(t),sin(t)>,conjugate=
> false);
> end:

normal_curvature(<(R+r*cos(u))*cos(v),
> (R+r*cos(u))*sin(v),r*sin(u)>,0,0,t);

plot(normal_curvature(subs({R=5,r=1},<(R+r*cos(u))*
> cos(v),(R+r*cos(u))*sin(v),r*sin(u)>),0,0,t),t=0..2*Pi);

plot(normal_curvature(<u,v,u*v>,0,0,t),t=0..2*Pi);

evalf(Pi/4);

maximize(normal_curvature(<u,v,u*v>,0,0,t),t=0..2*Pi,
> location=true);

rule_surf:=proc(beta,delta)
> simplify(beta+v*delta,symbolic);
> end:

plot3d(rule_surf(<cos(u)|sin(u)|0>,<0|1|1>),u=0..2*Pi,
> v=-1..1,scaling=constrained,shading=XY,orientation=
> [-24,63]);

plot3d(rule_surf(<u|0|0>,<0|1|u>),u=-1..1,v=-1..1,
> scaling=constrained,shading=XY,lightmodel=light2,
> orientation=[-74,71]);

plot3d(rule_surf(<cos(u)|sin(u)|u>,map(diff,<cos(u)|
> sin(u)|u>,u)),u=0..3.5*Pi,v=-4..4,scaling=constrained,shading
> =XY,lightmodel=light2,orientation=[-133,73],grid=[50,20]);

plot3d(rule_surf(<cos(u)|sin(u)|u>,map(diff,<cos(u)|
> sin(u)|u>,u)),u=0..3.5*Pi,v=0..4,scaling=constrained,shading
> =XY,lightmodel=light2,orientation=[-133,73],grid=[50,10]);

shape(rule_surf(<cos(u)|sin(u)|u>,map(diff,<cos(u)|
> sin(u)|u>,u)));

shape_matrix(rule_surf(<cos(u)|sin(u)|u>
> ,map(diff,<cos(u)|sin(u)|u>,u)));

Determinant(shape_matrix(rule_surf(<cos(u)|sin(u)|u>
> ,map(diff,<cos(u)|sin(u)|u>,u))));

plot3d(rule_surf(<0|0|sin(u)>,<cos(u)|sin(u)|0>),u=0..
> 2*Pi,v=-1..1,scaling=constrained,shading=XY,lightmodel
> =light2,orientation=[-27,54]);

plot3d(rule_surf(<cos(u)*cos(2*u)|sin(u)*cos(2*u)|
> sin(2*u)>,map(diff,<cos(u)*cos(2*u)|sin(u)*cos(2*u)
> |sin(2*u)>,u)),u=0..3.5*Pi,v=-2..2,scaling=constrained,
> shading=XY,lightmodel=light2,orientation=[38,43],
> grid=[40,20]);

cu:=tubeplot(convert(<cos(u)*cos(2*u)|sin(u)*cos(2*u)|
> sin(2*u)>,list),u=0..2*Pi,radius=0.015,color=black,
> numpoints=200,scaling=constrained):

sphere1:=plot3d(<cos(u)*cos(v)|sin(u)*cos(v)|sin(v)>,
> u=0..2*Pi,v=-Pi/2..Pi/2):

display({cu,sphere1},style=wireframe,scaling=
> constrained,shading=XYZ,lightmodel=light3,
> orientation=[38,43]);

display({cu,sphere1},scaling=constrained,
> shading=XYZ,lightmodel=light3,orientation=[38,43]);

with(plots):

with(plots):

fsolve(JacobiSN(u,k1)=1,u);

plot({JacobiSN(u,k1),JacobiCN(u,k1),JacobiDN(u,k1)},
> u=-1.854074677..3*1.854074677);

delau:=[-a*(1-epsilon)*(EllipticK(kk)+
> EllipticF(JacobiSN(u,kk),kk))-a*(1+epsilon)*
> (EllipticE(kk)+EllipticE(JacobiSN(u,kk),kk)),a*(1+epsilon)*
> JacobiDN(u,kk)*cos(v),a*(1+epsilon)*JacobiDN(u,kk)*sin(v)];

with(plots):

plotund:=proc(a,eps,theta,phi)
> local kk,uu;
> kk:=2*sqrt(eps)/(1+eps);
> uu:=fsolve(JacobiSN(u,kk)=1,u);
> plot3d(subs({u=u(t),v=v(t)},[-a*(1-eps)*(EllipticK(kk)+
> EllipticF(JacobiSN(u,kk),kk))-a*(1+eps)*(EllipticE(kk)+
> EllipticE(JacobiSN(u,kk),kk)),a*(1+eps)*JacobiDN(u,kk)*
> cos(v),a*(1+eps)*JacobiDN(u,kk)*sin(v)]),u=-uu..uu,v=0..2*Pi,
> scaling=constrained,axes=boxed,orientation=[theta,phi],
> shading=XY,lightmodel=light3);
> end:

plotund(1,0.7,-79,70);

plotund(1,0.3,-65,70);

EFG(subs(kk=2*sqrt(epsilon)/(1+epsilon),delau));

lmn(subs(kk=2*sqrt(epsilon)/(1+epsilon),delau));

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:

MK(subs(kk=2*sqrt(epsilon)/(1+epsilon),delau));

ccposR:=<EllipticE(-u,1/sqrt(2))|
> -1/sqrt(2)*sin(u)*cos(v)|-1/sqrt(2)*sin(u)*sin(v)>;

ccposL:=<EllipticE(u,1/sqrt(2))+2*EllipticE(1/sqrt(2))|
> 1/sqrt(2)*sin(u)*cos(v)|1/sqrt(2)*sin(u)*sin(v)>;

ccpos1:=plot3d(ccposR,u=-1..0,v=0..2*Pi):

ccpos2:=plot3d(ccposL,u=-1..0,v=0..2*Pi):

display({ccpos1,ccpos2},scaling=
> constrained,shading=z,lightmodel=light2,orientation=[88,60]);