Ophiuroid

OPHIUROID

Curve studied by Uhlhorn in 1809.
From the Greek ophis "snake" and oura "tail".

Polar equation:

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Cartesian equation:

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Rational circular cubic with a double point.

Like all rational circular cubics, the ophiuroids have three geometrical definitions. They are:
- the cissoids with pole O of a circle (C) passing by O and a line (D) the symmetric image about O of which passes by A, the point diametrically opposed to O (here, A(a, b) and (D): x = – a).

- the pedals of a parabola with respect to a point on the tangent to the vertex (here, the parabola with focus A and tangent at the vertex Oy).

- the inverses of a hyperbola with respect to one of its points (here, the hyperbola

, the reference circle being the circle (O, r)).

Note that the right ophiuroid is none other than the cissoid of Diocles.

Real ophiuroids!

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