## Hippopede of Proclus

Given the circle O(R) consider a point A inside it at distance OA = sR ( s in (0,1)). Then draw from O a parallel OD to each chord BC through
point A. Point D describes a curve called hypopede of Proclus. Its equation was found in Hippopede.html and is:
(x2+y2)2 - 4R2(x2 + (1-s2)y2) = 0.

The following properties are well known or can be easily proved [Mathcurve, Hippopede].
1) The curve is the envelope of circles whose centers H are on the ellipse with focus at A and its symmetric A w.r. to O. The circles are
also constrained to pass through the center of the ellipse. The major axis  of this ellipse is R and its minor axis is b = |AS| = sR,
determined by the position of point A inside the circle O(R).
2) The medial line BG of OD is tangent to the previous ellipse at a point H of it.
3) The tangent at D to the hippopede is the reflexion on BG of the tangent at O of the generating circle passing through D.
4) The inversion with respect to circle O(R) maps the ellipse onto the curve resulting from the hippopede by a π/2-rotation about O.
5) The Pedal curve of the ellipse with respect to its center O is the hippopede described by point G. This curve is homothetic to the
original hippopede by the ratio 1/2.

See the derivation of the equation in Hippopede.html [Mathcurve, Hippopede], [Seggern, p. 88], [Booth, II, p. 163].