Cissoid


In geometry, a cissoid is a plane curve generated from two given curves, and a point . Let be a variable line passing through and intersecting at and at. Let be the point on so that Then the locus of such points is defined to be the cissoid of the curves, relative to.
Slightly different but essentially equivalent definitions are used by different authors. For example, may be defined to be the point so that This is equivalent to the other definition if is replaced by its reflection through. Or may be defined as the midpoint of and ; this produces the curve generated by the previous curve scaled by a factor of 1/2.

Equations

If and are given in polar coordinates by and respectively, then the equation describes the cissoid of and 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, 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 and, which of these equations can be eliminated due to duplication.
For example, let and 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.
If each and are given by the parametric equations
and
then the cissoid relative to the origin is given by

Specific cases

When is a circle with center then the cissoid is conchoid of.
When and are parallel lines then the cissoid is a third line parallel to the given lines.

Hyperbolas

Let and be two non-parallel lines and let be the origin. Let the polar equations of and be
and
By rotation through angle we can assume that Then the cissoid of and 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.

Cissoids of Zahradnik

A cissoid of 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. Specifically: