Slicing Doughnuts

Torus-Cyan

It is well-known that an ellipse is the locus of all points such that the sum of their distances from two fixed points, the foci, is constant. Thus, a gardener may map out an elliptical flower-bed by driving two stakes into the ground, looping a rope around them and pulling it taut with a pointed stick, tracing out a curve on the ground.

Cassinian Ovals

There is another family of curves, Cassinian ovals, for which the product of the distances from two fixed points is constant. A Cassinian oval may be a simple convex curve like an ellipse, an indented oval, a figure-of-eight curve or a disjoint pair of ovals.

Fig. 1. Cassinian ovals. The red curve is the Lemniscate of Bernoulli.

Assuming that the foci are at ${(e,0)}$ and ${(-e,0)}$ and the product of the distances from foci to a point on the curve is ${a^2}$ , the equation of a Cassinian oval is easily found:

$\displaystyle (x^2+z^2)^2 - 2e^2(x^2-z^2) = a^4-e^4$

The shape of the curve depends on ${a/e}$ . When ${a<e}$ , the curve comprises two loops, each surrounding a focus. When ${a=e}$ , the curve is the lemniscate of Bernoulli, a figure-of-eight with a double point at the origin. When ${a > e}$ , the curve is a single loop enclosing both foci. It is indented or peanut-shaped when ${e < a < \sqrt{2}e}$ and convex when ${a \ge\sqrt{2}e}$ .

Giovanni Domenico Cassini (1625–1712) was an Italian mathematician, astronomer and engineer, and the oval curves studied by him were at one stage considered as possible solutions for planetary orbits.

Slicing a Doughnut

A torus, (Greek ${\mathit{\sigma\pi\epsilon\iota\rho\alpha}}$ : torus) may be considered as a product of two circles. Suppose ${(\rho,\theta,z)}$ are cylindrical coordinates with the axis of the torus on the ${z}$ -axis. Suppose we consider a circle in the ${xz}$ -plane, centred at ${x=a, z=0}$ with radius ${b}$ . We assume ${b<a}$ . If this circle is rotated about the ${z}$ -axis, a torus results.

A half-plane ${(\rho,z)}$ through the axis cuts the torus in a circle. Any position on the torus can be specified by giving the azimuthal angle ( ${\theta}$ ) of this plane (called the toroidal angle) and the position of the point on the circle (the poloidal angle ${\phi}$ ). We can write the equations of a torus in parametric form:

$\displaystyle \begin{array}{rcl} x &=& (a + b\cos\phi)\cos\theta \\ y &=& (a + b\cos\phi)\sin\theta \\ z &=& b\sin\phi \end{array}$

The spiric sections are the intersections of planes parallel to the axis of the torus. They are
of various forms depending upon the distance of the plane from the axis. Suppose we take the plane ${y=y_0}$ or, using the parametric equations ${(a + b\cos\phi)\sin\theta=y_0}$ . Substituting this into the other equations, we get a relationship between ${x}$ and ${z}$ :

$\displaystyle [(x^2+z^2)+(a^2-b^2+y_0^2)]^2 = 4a^2(x^2+y_0^2) \ \ \ \ \ \ (1)$

These curves are known as the spiric sections of Perseus. Almost nothing is known about this mathematician. We do not where he was born or where he lived, only that he was active in the second century BC. Proclus (AD 411–485) wrote that Perseus is associated with the spiric curves in the same way as Apollonius is associated with the conics.

Fig. 2. Spiric sections: Intersections between a torus and a plane parallel to its axis.

The general equation of a spiric section is

$\displaystyle (x^2+z^2)^2 = gx^2+hz^2+f\ \ \ \ \ \ (2)$

where ${f}$ , ${g}$ and ${h}$ are constants. We see that equation ( 1 ) above is of this form.

If ${g = -h = 2e^2}$ , equation (2) becomes

$\displaystyle (x^2+z^2)^2 - 2e^2(x^2 - z^2) = f$

which is a Cassinian oval. Thus, spiric sections include the ovals of Cassini as special cases. A set of such curves is shown in Figure 2 above.

Spiric sections arise in a variety of physical contexts. For example, the potential of a pair of line sources at positions ${\mathbf{r}_1}$ and ${\mathbf{r}_2}$ is

$\displaystyle \log|\mathbf{r}-\mathbf{r}_1| + \log|\mathbf{r}-\mathbf{r}_2|$

So, the isopotential lines are ${ |\mathbf{r}-\mathbf{r}_1|\cdot|\mathbf{r}-\mathbf{r}_2|}$ , which are Cassinian ovals.

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