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KAMPYLE OF EUDOXUS

From the Greek Kampulos: curved
Eudoxus of Cnidus (406 BC - 355 BC): Greek astronomer, mathematician and philosopher. Other name: Clairaut's curve. |

Polar equation:
(Clairaut's curve).
Cartesian equation: or (compare with that of Gerono's lemniscate). Rational quartic. Equation of the parabolic asymptotes: . |

If for a point P travelling on a circle (C)
of centre O, the tangent to C at P cuts Ox
in Q, then the Kampyle of Eudoxus is the locus of the intersection
point between the line (OP) and the line parallel to Oy passing
through Q. |

The Kampyle (in red, opposite) is also the locus of the
focus of a parabola constrained to stay tangent to a straight line at a
fixed point. Therefore, it is a glissette.
For a parabola of parameter |

The Kampyle of Eudoxus also is the radial curve of the catenary curve (here, up to rotation by an angle of p/2),

as well as the inverse of the double egg,

It was studied by Eudoxus because it is a duplicatrix.
Indeed, its intersection point *Q* with a circle of centre *C*
passing through *O* (equation
) is at distance
of *O*.

The Kampyle can also be found as the *rolling curve*
of the linear
conchoidal motion, and as the *base* of the Kappa
motion (see this page).

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© Robert FERRÉOL 2017