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

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.[br][br]The cissoid has a cusp at the point, and the tangent line is an asymptote.

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

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

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?

In this figure, be certain to drag points [i]Q[/i] and [i]A[/i].