Spiric section

File:Spiric section.png

a=1, b=2, c=0, 0.8, 1

A spiric section is a special case of a toric section, which is the intersection of a plane with a torus (σπειρα in ancient Greek). Spiric sections are toric sections in which the intersecting plane is parallel to the rotational symmetry axis of the torus. Spiric sections were discovered by the ancient Greek geometer Perseus in roughly 150 BC, and are assumed to be the first toric sections to be described.

Equations

Start with the usual equation for the torus:

$(x^{2}+y^{2}+z^{2}+b^{2}-a^{2})^{2}=4b^{2}(x^{2}+y^{2})$.

Interchanging y and z so that the axis of revolution is now on the xy-plane, and setting z=c to find the curve of intersection gives

$(x^{2}+y^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(x^{2}+c^{2})$.

In this formula, the torus is formed by rotating a circle of radius a with its center following another circle of radius b (not necessarily larger than a, self-intersection is permitted). The parameter c is the distance from the intersecting plane to the axis of revolution. There are no spiric sections with c > b + a, since there is no intersection; the plane is too far away from the torus to intersect it.

From this, a spiric section is a bicircular algebraic curves of degree 4 and symmetric with respect to both the x and y axes.

Expanding the equation gives a more symmetric form

$(x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f$

where

$d=2(a^{2}+b^{2}-c^{2}),\ e=2(a^{2}-b^{2}-c^{2}),\ f=-(a+b+c)(a+b-c)(a-b+c)(a-b-c)$.

In polar coordinates this becomes

$(r^{2}-a^{2}+b^{2}+c^{2})^{2}=4b^{2}(r^{2}\cos ^{2}\theta +c^{2})$

or

$r^{4}=dr^{2}\cos ^{2}\theta +er^{2}\sin ^{2}\theta +f$.

Examples of spiric sections

Examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli. The Cassini oval has the remarkable property that the product of distances to two foci are constant. For comparison, the sum is constant in ellipses, the difference is constant in hyperbolae and the ratio is constant in circles.

References

• MacTutor history
• MathWorld description
• 2Dcurves.com description
• "Spirique de Persée" at Encyclopédie des Formes Mathématiques Remarquables