Construct the torus using the command "tubeplot".

> restart;

> with(plots):

Warning, the name changecoords has been redefined

> with(plottools):

> vrml(c,"d:/9/8-1.wrl",background_color=white):

> c;

Construct two tori as thus:

> restart;

> with(plots):

Warning, the name changecoords has been redefined

> b:=tubeplot([2+2*cos(t),0,2*sin(t)],t=0..2*Pi,scaling=constrained,radius=0.6,grid=[30,30],color=green):

> c:=display(a,b):

> with(plottools):

> vrml(c,"d:/9/8-2.wrl",background_color=white):

> c;

Construct the torus using the command "plot3d".

> restart;

> r:=3+cos(t):

> x:=r*cos(s):y:=r*sin(s):z:=sin(t):

> c:=plot3d([x,y,z],s=0..2*Pi,t=0..2*Pi,scaling=constrained,color=blue):

> with(plottools):

> vrml(c,"d:/9/8-3.wrl",background_color=white):

> c;

Construct the torus by pasting two patches as thus:

> restart;

> with(plots):

Warning, the name changecoords has been redefined

> r:=3+cos(t):

> x:=r*cos(s):y:=r*sin(s):z:=sin(t):

> s:=u+v:

> t:=k*u:

> a:=plot3d([x,y,z],u=0..2*Pi,v=0..Pi/k,scaling=constrained,color=yellow,grid=[100,20]):

> b:=plot3d([x,y,z],u=0..2*Pi,v=Pi/k..2*Pi/k,scaling=constrained,color=blue,grid=[100,20]):

> c:=display(a,b):

> with(plottools):

> vrml(c,"d:/9/8-4.wrl",background_color=white):

> c;

Construct one half of a torus so its cross-section consists of two concentric circles.

> restart;

> r:=3+cos(t):

> x:=r*cos(s):y:=r*sin(s):z:=sin(t):

> c:=plot3d([x,y,z],s=0..2*Pi,t=0..Pi,scaling=constrained,color=blue):

> with(plottools):

> vrml(c,"d:/9/8-5.wrl",background_color=white):

> c;

Construct one half of a torus so its cross-section consists of two disjoint circles.

> restart;

> with(plots):

Warning, the name changecoords has been redefined

> with(plottools):

> vrml(c,"d:/9/8-6.wrl",background_color=white):

> c;

Construct one half of a torus so its cross-section consists of two intersecting circles.

> restart;

> a:=3:b:=4:c:=sqrt(a^2+b^2):

> r:=c+a*cos(t):

> x:=r*cos(s):y:=r*sin(s):z:=a*sin(t):

> u:=solve(b*z=a*x,s):

> c:=plot3d([x,y,z],s=-u..u,t=0..2*Pi,scaling=constrained,grid=[40,100],color=blue):

> with(plottools):

> vrml(c,"d:/9/8-7.wrl",background_color=white):

> c;

Construct one half of a torus so its cross-section consists of two interlocking circles in space.

> restart;

> x0:=5*cos(t):

> y0:=3+5*sin(t):z0:=0:

> s:=arccos(4/5):

> x1:=x0*cos(s)-z0*sin(s):y1:=y0:z1:=x0*sin(s)+z0*cos(s):

> x:=x1*cos(u)-y1*sin(u):y:=x1*sin(u)+y1*cos(u):z:=z1:

> c:=plot3d([x,y,z],t=0..2*Pi,u=0..Pi,scaling=constrained,color=blue):

> with(plottools):

> vrml(c,"d:/9/8-8.wrl",background_color=white):

> c;

Construct a knot as thus:

> restart;

> with(plots):

Warning, the name changecoords has been redefined

> x:=r*cos(s):y:=r*sin(s):z:=sin(5/2*s):

> c:=tubeplot([x,y,z],s=0..5*Pi,radius=0.7,grid=[100,20],scaling=constrained,color=blue):

> with(plottools):

> vrml(c,"d:/9/8-9.wrl",background_color=white):

> c;

Construct knots as thus:

> restart;

> with(plots):

Warning, the name changecoords has been redefined

> x:=r*cos(s):y:=r*sin(s):z:=sin(1/5*s):

> c:=tubeplot([x,y,z],s=0..10*Pi,radius=0.3,grid=[100,20],scaling=constrained,color=blue):

> with(plottools):

> vrml(c,"d:/9/8-10.wrl",background_color=white):

> c;

Construct a twisted torus whose sections are formed by regular triangles:

> restart;

> p:=2/3*Pi:

> r1:=3+cos(t/3):

> x1:=r1*cos(t):y1:=r1*sin(t):z1:=sin(t/3):

> r2:=3+cos(t/3+p):

> x2:=r2*cos(t):y2:=r2*sin(t):z2:=sin(t/3+p):

> x:=(1-s)*x1+s*x2:y:=(1-s)*y1+s*y2:z:=(1-s)*z1+s*z2:

> c:=plot3d([x,y,z],s=0..1,t=0..6*Pi,scaling=constrained,grid=[3,80],color=blue):

> with(plottools):

> vrml(c,"d:/9/8-11.wrl",background_color=white):

> c;

>