In geometry, a cissoid is a curve generated from two given curves C1, C2 and a point O. Let L be a variable line passing through O and intersecting C1 at P1 and C2 at P2. Let P be the point on L so that OP = P1P2. Then the locus of such points P is defined to be the cissoid of the curves C1, C2relative 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 = OP1 + OP2. This is equivalent to the other definition if C1 is replaced by its reflection through O. Or P may be defined as the midpoint of P1 and P2; this produces the curve generated by the previous curve scaled by a factor of 1/2. The word "cissoid" comes from the lit=ivy shaped from κισσός, 'ivy', and -οειδής, 'having the likeness of'.
Equations
If C1 and C2 are given in polar coordinates by and respectively, then the equation 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, letC1 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. If each C1 and C2 are given by the parametric equations and then the cissoid relative to the origin is given by
Specific cases
When C1 is a circle with center O then the cissoid is conchoid of C2. When C1 and C2 are parallel lines then the cissoid is a third line parallel to the given lines.
Hyperbolas
Let C1 and C2 be two non-parallel lines and let O be the origin. Let the polar equations of C1 and C2 be and By rotation through angle, we can assume that. 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.
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:
