Definition of the ellipse

The ellipse arises from a very simple geometric condition, which generalises that of the parabola: instead of a single focus, two are taken, and one requires that the sum of the distances from them be constant.

Definition — Ellipse

An ellipse is the locus of points $P$ of the plane such that the sum of the distances from two fixed points $F_1, F_2$ (called foci) is a positive constant, greater than the distance between the foci themselves: $P\in E \iff d(P,F_1) + d(P,F_2) = 2a, \qquad 2a > d(F_1,F_2).$

For every point $P$ of the ellipse we have $d(P,F_1)+d(P,F_2)=2a$; $a$ is the semi-major axis, $b$ the semi-minor axis, and the foci lie at a distance $c=\sqrt{a^2-b^2}$ from the origin.

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Topics: Ellipse
Concepts: Ellipse · Foci · Locus · Semi-axes