Toric section

Toric section

A toric section is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone.

Mathematical formulae

In general, toric sections are fourth-order (quartic) plane curves of the form

left(x^{2} + y^{2} right)^{2} + a x^{2} + b y^{2} + cx + dy + e = 0.

Spiric sections

A special case of a toric section is the spiric section (σπειρα = torus in ancient Greek), in which the intersecting plane is parallel to the rotational symmetry axis of the torus. They were discovered by the ancient Greek geometer Perseus in roughly 150 BC. Well-known examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli.

General toric sections

More complicated figures such as an annulus can be created when the intersecting plane is perpendicular or oblique to the rotational symmetry axis.

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