Supertoroid

Supertoroids with Template:Math, and different combinations for the parameters Template:Mvar and Template:Mvar.

In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surfaces (technically, a topological torus) whose shape is defined by mathematical formulas similar to those that define the superellipsoids. The plural of "supertorus" is either supertori or supertoruses.

The family was described and named by Alan Barr in 1994.

Barr's supertoroids have been fairly popular in computer graphics as a convenient model for many objects, such as smooth frames for rectangular things. One quarter of a supertoroid can provide a smooth and seamless 90-degree joint between two superquadric cylinders. However, they are not algebraic surfaces (except in special cases).

Formulas

Alan Barr's supertoroids are defined by parametric equations similar to the trigonometric equations of the torus, except that the sine and cosine terms are raised to arbitrary powers. Namely, the generic point Template:Math of the surface is given by $$P(u,v)=\left(\begin{array}{c}X(u,v)\\ Y(u,v)\\ Z(u,v)\end{array}\right)=\left(\begin{array}{c}(a+C_u^s)C_v^t\\ (b+C_u^s)S_v^t\\ S_u^s\end{array}\right)$$ where $$\begin{array}{cc}{C}_{\theta }^{\epsilon }& =\mathrm{sgn}(\cos \theta ){|\cos \theta |}^{\epsilon },\\ S_{\theta }^{\epsilon }& =\mathrm{sgn}(\sin \theta ){|\sin \theta |}^{\epsilon },\end{array}$$ Template:Math is the sign function, and the parameters Template:Mvar range from 0 to 360 degrees (0 to 2π radians).

In these formulas, the parameter Template:Math controls the "squareness" of the vertical sections, Template:Math controls the squareness of the horizontal sections, and Template:Math are the major radii in the Template:Mvar and Template:Mvar directions. With Template:Math and Template:Math one obtains the ordinary torus with major radius Template:Mvar and minor radius 1, with the center at the origin and rotational symmetry about the Template:Mvar-axis.

In general, the supertorus defined as above spans the intervals: $$\begin{array}{ccccc}-(a+1)& \le & x& \le & +(a+1)\\ -(b+1)& \le & y& \le & +(b+1)\\ -1& \le & z& \le & +1\end{array}$$ The whole shape is symmetric about the planes Template:Math, Template:Math, and Template:Math. The hole runs in the Template:Mvar direction and spans the intervals $$\begin{array}{ccccc}-(a-1)& \le & x& \le & +(a-1)\\ -(b-1)& \le & y& \le & +(b-1)\\ -\mathrm{\infty }& \le & z& \le & +\mathrm{\infty }\end{array}$$

A curve of constant Template:Mvar on this surface is a horizontal Lamé curve with exponent Template:Tmath scaled in Template:Mvar and Template:Mvar and displaced in Template:Mvar. A curve of constant Template:Mvar, projected on the plane Template:Math or Template:Math, is a Lamé curve with exponent Template:Tmath scaled and horizontally shifted. If Template:Math, the curve is planar and spans the intervals: $$\begin{array}{ccccc}a-1& \le & x& \le & a+1\\ -1& \le & z& \le & +1\end{array}$$ and similarly if Template:Math. The curve is also planar if Template:Math.

In general, if Template:Math and Template:Mvar is not a multiple of 90 degrees, the curve of constant Template:Mvar will not be planar; and, conversely, a vertical plane section of the supertorus will not be a Lamé curve.

The basic supertoroid shape defined above is often modified by non-uniform scaling to yield supertoroids of specific width, length, and vertical thickness.

Plotting code

The following GNU Octave code generates plots of a supertorus:

 function supertoroid(epsilon,a)
  n=50;
  d=.1;
  etamax=pi;
  etamin=-pi;
  wmax=pi;
  wmin=-pi;
  deta=(etamax-etamin)/n;
  dw=(wmax-wmin)/n;
  k=0;
  l=0;
  for i=1:n+1
    eta(i)=etamin+(i-1)*deta;
    for j=1:n+1
      w(j)=wmin+(j-1)*dw;
      x(i,j)=a(1)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(cos(w(j)))*abs(cos(w(j)))^epsilon(2);
      y(i,j)=a(2)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(sin(w(j)))*abs(sin(w(j)))^epsilon(2);
      z(i,j)=a(3)*sign(sin(eta(i)))*abs(sin(eta(i)))^epsilon(1);
    endfor;
  endfor;
   mesh(x,y,z);
 endfunction;

See also

• Superellipsoid
• Superegg
• Superquadric