Definition

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, and a point P is found in the line such that AP = QR, these lengths being measured in the direction indicated by the order of the letters. Then the locus of P is called the cissoid of S and S' with respect to A.

Thus the cissoid of two concentric circles, radii r1r2, with respect to their common centre is a circle with the same centre and radius |r1 –r2|.

The Cissoid of Diodes

This is the cissoid of a circle and a straight line touching it, with respect to the point on the circumference of the circle diametrically opposite to the point of contact. In Fig. 89, A is the fixed point, S and S' are the circle and the tangent at 5, and AP = QR.

This curve may be used for finding two mean proportionals between two given lengths. In Fig. 89, OU is the first of two mean proportionals between OC and OL; or, if the circle is of unit radius, the measure of OU is the cube root of that of OL. (Hint for proof : Let AO be a and let angle OAP be θ. Express coordinates of U,P and L in terms of a and θ.)

For other properties of this curve, see the summary given below.