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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 (G1) and (G2) with respect to a point O is the locus (G) of points M such that  where M1 is a point on (G1) and M2 is a point on (G2), with M1M2 and O aligned.

Therefore, the cissoid is the medial curve of pole O of the curves (G'1) and (G'2), the images of (G1) and (G2) 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 (G1) into its symmetrical image about O in the definition we adopted.

Examples:
 - when (G1) and (G2) are two parallel straight lines, the cissoid is a third parallel line.
 - when (G1) and (G2) are two secant straight lines, the cissoids are the hyperbolas passing through O, with asymptotes (G1) and (G2).

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 (G1).
 - when (G1) is a conic, (G2) is a line, and O is on the conic, we get the cissoids of Zahradnik.
 - when (G1) and (G2) are circles and O is on one of them, we get the rational bicircular quartics.
 - when (G1) and (G2) 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 (G1) and (G2) 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).
 
 
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© Robert FERRÉOL  2017