Editing Ellipse (section)

{{Short description|Plane curve}}
{{About|the geometric figure}}
{{distinguish|Ellipsis|Eclipse|Ecliptic}}
[[File:Ellipse-conic.svg|thumb|An ellipse (red) obtained as the intersection of a cone with an inclined plane.]]
[[File:Ellipse-var.svg|thumb|Ellipses: examples with increasing eccentricity]]

In [[mathematics]], an '''ellipse''' is a [[plane curve]] surrounding two [[focus (geometry)|focal points]], such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a [[circle]], which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its [[eccentricity (mathematics)|eccentricity]] <math>e</math>, a number ranging from <math>e = 0</math> (the [[Limiting case (mathematics)|limiting case]] of a circle) to <math>e = 1</math> (the limiting case of infinite elongation, no longer an ellipse but a [[parabola]]).

An ellipse has a simple [[algebra]]ic solution for its area, but for [[Perimeter of an ellipse|its perimeter]] (also known as [[circumference]]), [[Integral|integration]] is required to obtain an exact solution.

The largest and smallest [[diameter]]s of an ellipse, also known as its width and height, are typically denoted {{mvar|2a}} and {{mvar|2b}}. An ellipse has four [[extreme point]]s: two ''[[Vertex (geometry)|vertices]]'' at the endpoints of the [[major axis]] and two ''co-vertices'' at the endpoints of the minor axis.
[[File:Ellipse-def0.svg|300px|thumb|Notable points and line segments in an ellipse.]]

[[Analytic geometry|Analytically]], the equation of a standard ellipse centered at the origin is:
<math display="block">\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1 .</math>
Assuming <math>a \ge b</math>, the foci are <math>(\pm c, 0)</math> where <math display="inline">c = \sqrt{a^2-b^2}</math>, called [[#Linear_eccentricity|''linear eccentricity'']], is the distance from the center to a focus. The standard [[parametric equation]] is:
<math display="block">(x,y) = (a\cos(t),b\sin(t)) \quad \text{for} \quad 0\leq t\leq 2\pi.</math>

Ellipses are the [[closed curve|closed]] type of [[conic section]]: a plane curve tracing the intersection of a [[cone]] with a [[plane (mathematics)|plane]] (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and [[hyperbola]]s, both of which are [[open curve|open]] and [[unbounded set|unbounded]]. An angled [[Cross section (geometry)|cross section]] of a right circular [[cylinder (geometry)#Cylindric section|cylinder]] is also an ellipse.

An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the [[#Eccentricity and the directrix property|''directrix'']]: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant, called the [[#Eccentricity and the directrix property|''eccentricity'']]:
<math display="block">e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}.</math>

Ellipses are common in [[physics]], [[astronomy]] and [[engineering]]. For example, the [[Kepler orbit|orbit]] of each planet in the [[Solar System]] is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the [[Barycentric coordinates (astronomy)|barycenter]] of the Sun{{ndash}}planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by [[ellipsoid]]s. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under [[parallel projection|parallel]] or [[perspective projection]]. The ellipse is also the simplest [[Lissajous figure]] formed when the horizontal and vertical motions are [[Sine wave|sinusoid]]s with the same frequency: a similar effect leads to [[elliptical polarization]] of light in [[optics]].

The name, {{lang|grc|ἔλλειψις}} ({{Transliteration|grc|élleipsis}}, "omission"), was given by [[Apollonius of Perga]] in his ''Conics''.