CISSOID OF TWO CURVES

 From the Greek Kissos: ivy.

 Polar equation of the cissoid of pole O of the curves  and :  .

The cissoid of two curves  and  with respect to a point O is the locus  of points M such that  where M1 is a point on  and M2 is a point on , with M1M2 and O aligned.

Therefore, the cissoid is the medial curve of pole O of the curves  and , the images of  and  by the homothety of centre O and ration 1/2.

In dotted lines, the curves (G1) and (G2), in blue, the curves (G'1) and (G'2) the medial of which is the cissoid.
Remark: the cissoid of the cissoid and the symmetrical image of one of the initial curves with respect to O is the other initial curve.

Sometimes, the cissoid is defined as the locus of points M such that ; this amounts, of course, to changing  into its symmetrical image about O in the definition we adopted.

Examples:
- when  and  are two parallel straight lines, the cissoid is a third parallel line.
- when  and  are two secant straight lines, the cissoids are the hyperbolas passing through O, with asymptotes  and .

In mauve, the two lines, and in blue their homothetic image, of which the hyperbola is the medial curve.
- when (G2) is a circle a and O is its centre, we get the conchoids of the curve .
- when  is a conic,  is a line, and O is on the conic, we get the cissoids of Zahradnik.
- when  and  are circles and O is on one of them, we get the rational bicircular quartics.
- when  and  are circles and O is in the middle of the two centres, we get the Booth curves, of which the lemniscate of Bernoulli is an example.

- The parabolic folium is the cissoid of a line and a semicubical parabola.

- the beetle curves are the cissoids of a circle and a four-leaved rose.

Remark: when the two curves  and  coincide, the cissoid is composed of the image of it by the homothety of centre O and ratio 2, but also, possibly, of another part (because the points M1 and M2 can be different).