Ellipse

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In mathematics, an ellipse is an example of a conic section. It can be defined as the locus of all points in a plane that have the same sum of distances from two given fixed points (called foci) of the plane. If the two foci coincide, then the ellipse is a circle, which is a degenerate case to which some of the following does not apply.

The line passing through the foci is called the major axis of the ellipse. The line passing through the centre of the ellipse (the midpoint of the foci) at right angles to the major axis is called the minor axis. The size and shape of an ellipse are determined by two constants, conventionally denoted a and b. a is the length of the semi-major axis (the distance from the centre of the ellipse to the ellipse itself, measured along the major axis). Note that the sum of distances from the foci for any point on the ellipse is 2a. b is the length of the semi-minor axis (the distance from the centre of the ellipse to the ellipse itself, measured along the minor axis). An ellipse centred at the origin with its major axis along the x-axis is defined by the equation

  x2    y2
 --- + --- = 1.
  a2    b2

The same ellipse is also represented by the parametric equations x = a cos t and y = b sin t.

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e, which is related to a and b by the formula b2 = a2(1 - e2). The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The distance between the foci is 2ae.

The semi-latus rectum of an ellipse, usually denoted l (a lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to a and b by the formula al = b2. An ellipse with one focus at the origin and the other on the negative x-axis is given by the equation

r (1 + e cos θ) = l

in polar coordinates.

An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

The area enclosed by an ellipse is πab, where π is Archimedes' constant. The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind.