Ovale de Cassini

In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2. Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in the late 17th century. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini. A Cassini oval is a set of points, such that for any point of the set, the product of the distances to two fixed points is a constant, usually written as where : As with an ellipse, the fixed points are called the foci of the Cassini oval. If the foci are (a, 0) and (−a, 0), then the equation of the curve is When expanded this becomes The equivalent polar equation is The curve depends, up to similarity, on e = b/a. When e < 1, the curve consists of two disconnected loops, each of which contains a focus. When e = 1, the curve is the lemniscate of Bernoulli having the shape of a sideways figure eight with a double point (specifically, a crunode) at the origin. When e > 1, the curve is a single, connected loop enclosing both foci. It is peanut-shaped for and convex for . The limiting case of a → 0 (hence e → ), in which case the foci coincide with each other, is a circle. The curve always has x-intercepts at ± c where c2 = a2 + b2. When e < 1 there are two additional real x-intercepts and when e > 1 there are two real y-intercepts, all other x- and y-intercepts being imaginary. The curve has double points at the circular points at infinity, in other words the curve is bicircular. These points are biflecnodes, meaning that the curve has two distinct tangents at these points and each branch of the curve has a point of inflection there.