Hippopede

Quartic from chords through a point

From a fixed point A draw chords BC of a fixed circle. Find the locus of points D, such that OD is parallel and
equal to the chord BC.

This is the Hippopede of Proclus curve [Mathcurve, http://www.mathcurve.com/courbes2d/booth/booth.shtml],
[Seggern, p. 88], [Booth, II, p. 163].
Considering arbitrary values for the parameter in the last equation and taking in particular s=sqrt(2) we obtain the
lemniscus of Bernoulli [Lockwood, p. 111].
Another view of this curve and some additional remarks are presented in Hippopede2.html .

Problem The figure shows an isosceles trapezium BCED which varies as BC changes its position. Its peculiarity
is that it has basis DE of length twice the length of BC. The hippopede can be considered as the path described by
vertices {D,E} as the trapezium changes position by requiring:
a) that BC passes through the fixed A,
b) that DE passes through the fixed O,
c) that DE has double the length of BC.
What about an isosceles trapezium varies satisfying conditions {a,b,c} but allowing in (c) more general than k=2
for the ratio k=DE/BC? The interesting cases occur when O is not the center of DE. In fact, it is easily seen that
for this particular case the resulting curve is homothetic to an hippopede. A figure illustrating the general case in
which point O is not the middle of DE can be seen in Hippopede_Gen.html .

Bibliography

[Booth] James Booth A treatise on some New Geometrical Methods 2 vols. Longmans, London 1877
[Lockwood] Lockwood, E. H. A Book of Curves Cambridge, Cambridge University Press., 1961
[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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Quartic from chords through a point

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Bibliography