In geometry, a cissoid (() is a plane curve generated from two given curves , and a point (the pole). Let be a variable line passing through and intersecting at and at . Let be the point on so that (There are actually two such points but is chosen so that is in the same direction from as is from .) 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.