# Cissoids

- Author:
- Steve Phelps

Let

*S*and*S'*be any two curves and let*A*be a fixed point. A straight line is drawn through*A*cutting*S*and*S'*at*Q*and*R*respectively. Point*P*is found on the line such that*AP*=*QR*, these lengths being measured in the direction indicated by the order of the labels. The locus of*P*is called the*. In the dynamic figures below, point***cissoid of S and S' with respect to A***Q*is typically**not**draggable. Of course, there may be an exception or two.## The Cissoid of Diocles

This is the cissoid of a circle and a line tangent to the circle with respect to a point on the circle diametrically opposed to the point of tangency.
The cissoid has a cusp at the point, and the tangent line is an asymptote.

## The Oblique Cissoid

The cissoid of a circle and a tangent line with respect to a point not diametrically opposed to the point of tangency.
The cissoid has a cusp at the point, and the tangent line is an asymptote, but the cissoid now crossed the tangent line.

## The Cissoid of a Circle and a Line not Tangent.

If the line passes through the center of the circle, it is a strophoid.

## The Cissoid of Two Intersecting Lines

The cissoid of two intersecting lines with respect to a point on on either of them is a curve with asymptotes parallel to the given curves. Does it look like a hyperbola?

## The Cissoid of a Parabola and its Directrix with Respect to the Focus

In this figure, be certain to drag points

*Q*and*A*.