Cissoid
If "C"1 and "C"2 are given in polar coordinates by formula_1 and formula_2 respectively, then the equation formula_3 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 
formula_4. So the cissoid is actually the union of the curves given by the equations 
formula_5
formula_6. 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
formula_7. The first branch of the cissoid is given by 
formula_8,
which is simply the origin. The ellipse is also given by 
formula_9,
so a second branch of the cissoid is given by
formula_10
which is an oval shaped curve. If each "C"1 and "C"2 are given by the parametric equations
formula_11
and
formula_12,
then the cissoid relative to the origin is given by 
formula_13. 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 
formula_14
and
formula_15.
By rotation through angle formula_16, we can assume that formula_17. Then the cissoid of "C"1 and "C"2 relative to the origin is given by
formula_18
formula_19
formula_20. Combining constants gives
formula_21
which in Cartesian coordinates is
formula_22. 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:
formula_23
is the cissoid of the circle formula_24 and the line formula_25 relative to the origin. formula_26
is the cissoid of the circle formula_24 and the line formula_28 relative to the origin. formula_29
is the cissoid of the circle formula_24 and the line formula_31 relative to the origin. This is, in fact, the curve for which the family is named and some authors refer to this as simply as cissoid. formula_34
is the cissoid of the ellipse formula_35 and the line formula_36 relative to the origin. To see this, note that the line can be written
formula_37
and the ellipse can be written
formula_38. So the cissoid is given by
formula_39
which is a parametric form of the folium.
Equations
Then the locus of such points "P" is defined to be the cissoid of the curves "C"1, "C"2 relative to "O".