Sample Maple Worksheet to create a torus and a sample path

restart;

with(plots):

f:=cos(beta)*(cos(alpha)+2);

g:=sin(beta)*(cos(alpha)+2);

h:=sin(alpha);

plot3d([f,g,h],alpha=0..Pi,beta=0..Pi,scaling=constrained,style=patchnogrid,shading=ZHUE);

Next we put a path on the torus. This will be a parametric curve (or parametric curves), depending only on one variable

The first path goes around the (great) cricle. In this case, alpha=0 and beta = t*Pi, t=0..1

Path1Functions:=subs({alpha=0, beta=t*Pi},f), subs({alpha=0, beta=t*Pi},g), subs({alpha=0, beta=t*Pi},h);

Path1:=spacecurve([Path1Functions, t=0..1],color=black,thickness=5):

Torus1:=plot3d([f,g,h],alpha=0..2*Pi,beta=0..2*Pi,scaling=constrained,style=patchnogrid,shading=ZHUE):

display3d({Torus1, Path1});

An example of a path that needs to be defined in pieces: In (beta, alpha) we go from (0,0) to (Pi, 0) to (Pi, Pi), to (Pi, 0)

Path2AF:=subs({beta=0,alpha=Pi*t},f), subs({beta=0,alpha=Pi*t},g), subs({beta=0, alpha=Pi*t},h);

Path2BF:=subs({beta=Pi*t,alpha=Pi},f), subs({beta=Pi*t,alpha=Pi},g), subs({beta=Pi*t, alpha=Pi},h);

Path2CF:=subs({beta=Pi,alpha=Pi*t},f), subs({beta=Pi,alpha=Pi*t},g), subs({beta=Pi, alpha=Pi*t},h);

Path2A:=spacecurve([Path2AF,t=0..1],color=black, thickness=5):

Path2B:=spacecurve([Path2BF,t=0..1],color=black, thickness=5):

Path2C:=spacecurve([Path2CF,t=0..1],color=black, thickness=5):

display3d({Torus1, Path2A, Path2B, Path2C});