- 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 cissoid of S and S' with respect to A. In the dynamic figures below, point 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.