Hippopede2

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].

Bibliography

[Booth] James Booth A treatise on some New Geometrical Methods 2 vols. Longmans, London 1877
[Mathcurve] Robert Ferreol Curves remarquables Encyclopedie des formes mathematiques remarquables
[Seggern] David von Seggern Curves and Surfaces CRC Boca Raton 1990

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Hippopede of Proclus

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