In mathematics, an ellipse (from the Greek ἔλλειψις, literally absence) is a conic section, the locus of points in a plane such that the sum of the distances to two fixed points is a constant. The two fixed points are called foci (singular- focus). An alternate definition would be that an ellipse is the path traced out by a point whose distance from a fixed point, called the focus, maintains a constant ratio less than one with its distance from a straight line not passing through the focus, called the directrix.

Overview

An ellipse is a type of conic section: if a conical surface is cut by a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres.

Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form

$A\; x^2\; +\; B\; xy\; +\; C\; y^2\; +\; D\; x\; +\; E\; y\; +\; F\; =\; 0\; \backslash ,$

such that , where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists.

The line segment AB, that passes through the foci and terminates on the ellipse, is called the major axis. The major axis is the longest segment that can be obtained by joining two points on the ellipse. The line segment CD, which passes through the center (halfway between the foci), perpendicular to the major axis, and terminates on the ellipse, is called the minor axis. The semimajor axis (denoted by a in the figure) is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis (denoted by b in the figure) is one half the minor axis.

If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero.

An ellipse centered at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix $A\; =\; PDP^T$, $D$ being a diagonal matrix with the eigenvalues of $A$, both of which are real positive, along the main diagonal, and $P$ being a real unitary matrix having as columns the eigenvectors of $A$. Then the axes of the ellipse will lie along the eigenvectors of $A$, and the (1 over the square root of the) eigenvalues are the lengths of the semimajor and semiminor axes, which are one-half of the lengths of the major and minor axes respectively.

An ellipse can be produced by multiplying the x coordinates of all points on a circle by a constant, without changing the y coordinates. This is equivalent to stretching the circle out in the x-direction.