Summer Session 8
July 27

Construct the torus using the command "tubeplot".

Construct two tori as thus:

Construct the torus using the command "plot3d".

Construct the torus by pasting two patches as thus:

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

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

Construct one half of a torus so its cross-section consists of two intersecting circles.
These circles are called Villarceaux circles. Reference: Z.A. Melzak, Invitation to Geometry, pp. 63-68.

Construct one half of a torus so its cross-section consists of two interlocking circles in space.
> restart;
> a:=3:b:=4:c:=sqrt(a^2+b^2);
> x0:=c*cos(t);
> y0:=a+c*sin(t);
> z0:=0;
> s:=arccos(b/c);
> 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;
> plot3d([x,y,z],t=0..2*Pi,u=0..Pi,scaling=constrained);

Construct a knot as thus:

Construct knots as thus:

Construct a twisted torus whose sections are formed by regular triangles:
> restart;p:=2*Pi/3;
> 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;
> plot3d([x,y,z],s=0..1,t=0..6*Pi,scaling=constrained,grid=[5,100]):

Construct the VRML files associated with all models in this exercise.
Location of the Cosmo Player:
http://poncelet.math.nthu.edu.tw/software/cosmoplayerinstall.exe