Ellipse

An ellipse is the set of all points in a plane such that the sum of the distances to two fixed points, called the foci, is constant. This constant equals 2a, where a is the length of the semi-major axis. The foci lie on the major axis on opposite sides of the center, at distance c from the center, with c^2 = a^2 − b^2, where b is the semi-minor axis. The eccentricity e is c/a, a value between 0 and 1, with e = 0 corresponding to a circle.

In a Cartesian coordinate system with the center at the origin and the major axis along the x-axis, an ellipse is described by the equation x^2/a^2 + y^2/b^2 = 1. If the ellipse is rotated, the equation may involve an xy term and, in general, can be written as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 with B ≠ 0 for rotated cases.

The ellipse’s vertices are at (±a, 0) and the co-vertices at (0, ±b). The area enclosed is πab. There is no simple exact formula for the circumference; a common approximation is Ramanujan’s formula P ≈ π[3(a + b) − sqrt((3a + b)(a + 3b))].

Applications of ellipses include celestial mechanics, where planetary orbits are ellipses with the sun at one focus; optics, where elliptical mirrors reflect light from one focus to the other; and various engineering and architectural contexts that exploit their reflective properties.

The term ellipse derives from the Greek elleipsis, meaning “falling short.”