- Spiric section
A spiric section is a special case of a
toric section , which is the intersection of a plane with atorus (σπειρα in ancient Greek). Spiric sections are toric sections in which the intersecting plane is parallel to the rotational symmetry axis of thetorus . Spiric sections were discovered by the ancient Greek geometer Perseus in roughly150 BC , and are assumed to be the first toric sections to be described.Mathematical description
In general, spiric sections are fourth-order (
quartic )plane curve swith three parameters:left( r^{2} - a^{2} + c^{2} + x^{2} + y^{2} ight)^{2} = 4r^{2} left( x^{2} + c^{2} ight)
In this formula, the
torus is formed by rotating a circle of radius a with its center following another circle of radius r (not necessarily larger than a, self-intersection is OK). The parameter c is the shortest distance from the intersecting plane to the (parallel) rotational symmetry axis. There are no spiric sections with c > r + a, since there is no intersection; the plane is too far away from the torus to intersect it.The overall scale dependence can be eliminated by setting r=1. Such normal-form spiric sections are a 2-parameter family of
quartic plane curve s.Examples of spiric sections
Well-known examples include the
hippopede and theCassini oval and their relatives, such as thelemniscate of Bernoulli . TheCassini oval has the remarkable property that the "product" of distances to two foci are constant. For comparison, the sum is constant inellipse s, the difference is constant inhyperbola e and the ratio is constant incircle s.External links
* [http://www-groups.dcs.st-and.ac.uk/~history/Curves/Spiric.html MacTutor history]
* [http://mathworld.wolfram.com/SpiricSection.html MathWorld description]
* [http://www.2dcurves.com/quartic/quartics.html 2Dcurves.com description]
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