Cartesian oval

In geometry, a Cartesian oval, named after René Descartes, is a plane curve, the set of points that have the same linear combination of distances from two fixed points.

Let $P$ and $Q$ be fixed points in the plane, and let $d(P, S)$ and $d(Q, S)$ denote the Euclidean distances from these points to a third variable point $S$ . Let $m$ and $a$ be arbitrary real numbers. Then the Cartesian oval is the locus of points S satisfying $d(P, S) + m d(Q, S) = a$ . The two ovals formed by the four equations $d(P, S) + m d(Q, S) = \pm a$ and $d(P, S) - m d(Q, S) = \pm a$ are closely related; together they form a quartic plane curve called the ovals of Descartes.

In the equation $d(P, S) + m d(Q, S) = a$ , when $m = 1$ and $a > d(P, Q)$ the resulting shape is an ellipse. In the limiting case in which P and Q coincide, the ellipse becomes a circle. When $m=a/{\text{d}}(P,Q)$ it is a limaçon of Pascal. If $m=-1$ and $0<a<{\text{d}}(P,Q)$ the equation gives a branch of a hyperbola and thus is not a closed oval.

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