Ellipse

ellipticalellipticeccentricityellipsoidellipsoidalsemi-ellipseelipticalEllipsesorbital area(pins and) string method
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.wikipedia
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Circle

circularcircles360 degrees
As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

Hyperbola

hyperbolicrectangular hyperbolahyperbolas
Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded.
(The other conic sections are the parabola and the ellipse.

Focus (geometry)

focifocusfocal points
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola.

Kepler orbit

Keplerian orbitKeplerianKeplerian ellipse
For example, the 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 barycenter of the Sun–planet pair).
In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space.

Apollonius of Perga

ApolloniusApollonius of PergeApollonian
The name, ἔλλειψις (, "omission"), was given by Apollonius of Perga in his Conics.
His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.

Lissajous curve

Lissajous figureLissajous figuresBowditch curve
The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.
. For a ratio of 1, the figure is an ellipse, with special cases including circles (

Limiting case (mathematics)

limiting caselimitinglimiting cases
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).

Elliptical polarization

elliptically polarizedellipticalpolarization ellipse
The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation.

Solar System

outer Solar Systeminner Solar Systemouter planets
For example, the 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 barycenter of the Sun–planet pair).
Following Kepler's laws, each object travels along an ellipse with the Sun at one focus.

Conic section

conicconic sectionsconics
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure).
2. Circle and ellipse

Cross section (geometry)

cross sectioncross-sectioncross sections
An angled cross section of a cylinder is also an ellipse.
The conic sections – circles, ellipses, parabolas, and hyperbolas – are plane sections of a cone with the cutting planes at various different angles, as seen in the diagram at left.

Whispering gallery

whispering galleriesThe Whispering GalleryWhispering Wa
This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery).
A whispering gallery is usually a circular, hemispherical, elliptical or ellipsoidal enclosure, often beneath a dome or a vault, in which whispers can be heard clearly in other parts of the gallery.

Parabola

parabolicparabolic curveparabolic arc
Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. 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).
This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram.

Eccentric anomaly

The parameter t (called the eccentric anomaly in astronomy) is not the angle of (x(t),y(t)) with the x-axis, but has a geometric meaning due to Philippe de La Hire (see Drawing ellipses below).
In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.

Trammel of Archimedes

ellipsographDo nothing grinderelliptic trammel
However, technical tools (ellipsographs) to draw an ellipse without a computer exist.
A trammel of Archimedes is a mechanism that generates the shape of an ellipse.

Director circle

Fermat–Apollonius circle
This circle is called orthoptic or director circle of the ellipse (not to be confused with the circular directrix defined above).
In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.

Confocal conic sections

confocal ellipsesconfocal ellipseconfocal quadrics
A similar method for drawing [[Confocal conic sections#Graves's theorem: the construction of confocal ellipses by a string|confocal ellipses]] with a closed string is due to the Irish bishop Charles Graves.
Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas.

True anomaly

The angle \theta in these formulas is called the true anomaly of the point.
It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse (the point around which the object orbits).

Matrix (mathematics)

matrixmatricesmatrix theory
An affine transformation of the Euclidean plane has the form, where A is a regular matrix (with non-zero determinant) and \vec f\!_0 is an arbitrary vector.

Dandelin spheres

Using Dandelin spheres, one can prove that any plane section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.
The first theorem is that a closed conic section (i.e. an ellipse) is the locus of points such that the sum of the distances to two fixed points (the foci) is constant.

Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is:

Elliptic integral

elliptic integralscomplete elliptic integral of the first kindcomplete elliptic integral of the second kind
where again a is the length of the semi-major axis, is the eccentricity, and the function E is the complete elliptic integral of the second kind,
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

Optics

opticalopticoptical system
The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.
In all other cases, where the two components either do not have the same amplitudes and/or their phase difference is neither zero nor a multiple of 90°, the polarization is called elliptical polarization because the electric vector traces out an ellipse in the plane (the polarization ellipse).

Rytz's construction

Rytz’s axis construction
With help of Rytz's construction the axes and semi-axes can be retrieved.
The Rytz’s axis construction is a basic method of descriptive Geometry to find the axes, the semi-major axis and semi-minor axis and the vertices of an ellipse, starting from two conjugated half-diameters.

Parallel projection

parallelparallel linear projectionprojection
A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection.