Cissoid - Equations

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, let C₁ 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

.