ellipse

Also known as ⬭

thumb|An ellipse (red) obtained as the intersection of a cone with an inclined plane. thumb|Ellipses: examples with increasing eccentricity

AI overview

An ellipse is an oval-shaped curve that you can create by slicing through a cone at an angle, and it's one of the fundamental shapes in geometry and astronomy. Ellipses matter because they describe the paths that planets and satellites follow as they orbit, making them essential to understanding how the universe works.

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Wikidata facts

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Commons category: Ellipses
Commons gallery: Ellipses
Unicode character: ⬭

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~64 min read

Article

62 sections

Contents

Definition as locus of points
In Cartesian coordinates
Standard equation
Parameters
Principal axes
Linear eccentricity
Eccentricity
Semi-latus rectum
Tangent
Shifted ellipse
General ellipse
Parametric representation
Standard parametric representation
Rational representation
Tangent slope as parameter
General ellipse
Polar forms
Polar form relative to center
Polar form relative to focus
Eccentricity and the directrix property
Focus-to-focus reflection property
Conjugate diameters
Definition of conjugate diameters
Theorem of Apollonios on conjugate diameters
Orthogonal tangents
Drawing ellipses
de La Hire's point construction
Pins-and-string method
Paper strip methods
Approximation by osculating circles
Steiner generation
As hypotrochoid
Inscribed angles and three-point form
Circles
Inscribed angle theorem for circles
Three-point form of circle equation
Ellipses
Inscribed angle theorem for ellipses
Three-point form of ellipse equation
Pole-polar relation
Metric properties
Area
Circumference
Arc length
Curvature
In triangle geometry
As plane sections of quadrics
Applications
Physics
Elliptical reflectors and acoustics
Planetary orbits
Harmonic oscillators
Phase visualization
Elliptical gears
Optics
Statistics and finance
Computer graphics
Optimization theory
See also
Notes
References
External links

thumb|An ellipse (red) obtained as the intersection of a cone with an inclined plane. thumb|Ellipses: examples with increasing eccentricity

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of both distances to the two 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 e, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola).