ellipses

An ellipse is a plane curve defined as the set of all points for which the sum of the distances to two fixed points, called foci, is constant. Equivalently, it is a conic section formed by intersecting a cone with a plane that cuts through both nappes at an angle less than the cone’s side.

In the standard orientation, with the major axis on the x-axis and the center at the origin, the ellipse is described by the equation x^2/a^2 + y^2/b^2 = 1, where a ≥ b > 0. The major axis length is 2a and the minor axis length is 2b. The foci lie at (±c, 0) where c^2 = a^2 − b^2, and the eccentricity e = c/a satisfies 0 ≤ e < 1. If a = b, the ellipse becomes a circle of radius a.

Key properties include symmetry about its major and minor axes and a constant sum of distances to the foci. The area of an ellipse is πab. Perimeter does not have a simple exact formula in elementary terms, though good approximations exist.

Ellipses can be rotated or tilted; in Cartesian coordinates their general form is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 with B^2 − 4AC < 0, which describes an ellipse in any orientation.

Applications of ellipses are widespread, including planetary orbits around stars, optical reflectors that focus light, and architectural or mechanical designs where their reflective or aesthetic properties are advantageous.

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