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CAPRICORNOID



k < 1

k > 1

 
Curve studied by Poncelet in 1862 (applications d'analyse et de géométrie, p. 460)
Name given by Poncelet by analogy with horns of goat (capra in Latin).
Ref : Loria page 244 and 788.

 
Cartesian parametrization: 
Polar equation:  where .
Cartesian equation: .
Circularrational quartic.

 
Given a fixed point A(a, 0) and a circle (C) of center O and radius ka, the capricornoid  is the place of the end M from a right angle POM, the point P describing the circle (C), and the point M belonging to the line (AP).

Note: if we exchange the roles of A and O, we obtain the curve of Jerabek, which has a similar shape for k > 1.

The transformation  is an involution by construction. Analytically defined by , it is a quadratic transformation like the Newton transformation.
k > 1
k < 1
Picture by Gino Loria

When A is inside the circle (k > 1), the curve is closed, included in the circle, and has a double crunodal point (A) and a double tacnodal point (O).

When A is on the circle (k = 1), we Gets the right trophoid :  (construction by Pierre Apel).

When A is outside (k < 1), there is a double crunodal point and a double tacnodal point, and two asymptotes.
 
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