Cissoid

In geometry, a cissoid (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped') is a plane curve generated from two given curves C1, C2 and a point O (the pole).

Then the locus of such points P is defined to be the cissoid of the curves C1, C2 relative to O.

Slightly different but essentially equivalent definitions are used by different authors.

This is equivalent to the other definition if C1 is replaced by its reflection through O.

describes the cissoid of C1 and C2 relative to the origin.

However, because a point may be represented in multiple ways in polar coordinates, there may be other branches of the cissoid which have a different equation.

Specifically, C1 is also given by So the cissoid is actually the union of the curves given by the equations It can be determined on an individual basis depending on the periods of f1 and f2, which of these equations can be eliminated due to duplication.

For example, let C1 and C2 both be the ellipse The first branch of the cissoid is given by which is simply the origin.

The ellipse is also given by so a second branch of the cissoid is given by which is an oval shaped curve.

Let the polar equations of C1 and C2 be and By rotation through angle

Then the cissoid of C1 and C2 relative to the origin is given by Combining constants gives which in Cartesian coordinates is This is a hyperbola passing through the origin.

So the cissoid of two non-parallel lines is a hyperbola containing the pole.

A similar derivation show that, conversely, any hyperbola is the cissoid of two non-parallel lines relative to any point on it.

A cissoid of Zahradnik (named after Karel Zahradnik) is defined as the cissoid of a conic section and a line relative to any point on the conic.

This is a broad family of rational cubic curves containing several well-known examples.