In geometry, a cissoid is a curve generated from two given curves C_{1}, C_{2} and a point O (the pole). Let L be a variable line passing through O and intersecting C_{1} at P_{1} and C_{2} at P_{2}. Let P be the point on L so that OP = P_{1}P_{2}. (There are actually two such points but P is chosen so that P is in the same direction from O as P_{2} is from P_{1}.) Then the locus of such points P is defined to be the cissoid of the curves C_{1}, C_{2} relative to O.
Slightly different but essentially equivalent definitions are used by different authors. For example, P may be defined to be the point so that OP = OP_{1} + OP_{2}. This is equivalent to the other definition if C_{1} is replaced by its reflection through O. Or P may be defined as the midpoint of P_{1} and P_{2}; this produces the curve generated by the previous curve scaled by a factor of 1/2.
The word "cissoid" comes from the Greek: κισσοειδής, lit. 'ivy shaped' from κισσός, 'ivy', and -οειδής, 'having the likeness of'.
If C_{1} and C_{2} are given in polar coordinates by and respectively, then the equation describes the cissoid of C_{1} and C_{2} 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, C_{1} 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 f_{1} and f_{2}, which of these equations can be eliminated due to duplication.
For example, let C_{1} and C_{2} 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 C_{1} and C_{2} are given by the parametric equations
and
then the cissoid relative to the origin is given by
When C_{1} is a circle with center O then the cissoid is conchoid of C_{2}.
When C_{1} and C_{2} are parallel lines then the cissoid is a third line parallel to the given lines.
Let C_{1} and C_{2} be two non-parallel lines and let O be the origin. Let the polar equations of C_{1} and C_{2} be
and
By rotation through angle , we can assume that . Then the cissoid of C_{1} and C_{2} 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. Specifically: