Ellipse

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, a number ranging from to .
An ellipse has a simple algebraic formula for its area, but for its perimeter, integration is required to obtain an exact solution.
The largest and smallest diameters of an ellipse, also known as its width and height, are typically denoted and. An ellipse has four extreme points: two vertices at the endpoints of the major axis and two co-vertices at the endpoints of the minor axis.
Analytically, the equation of a standard ellipse centered at the origin is:
Assuming, the foci are, where is the distance from the center to a focus. The standard parametrization is:
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane. Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a right circular 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 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:
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point. 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 ellipsoids. 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. 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.
The name, ἔλλειψις, was given by Apollonius of Perga in his Conics.

Definition as locus of points

An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane:
The midpoint of the line segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis, and the line perpendicular to it through the center is the minor axis. The major axis intersects the ellipse at two vertices, which have distance to the center. The distance of the foci to the center is called the focal distance or linear eccentricity. The quotient is defined as the eccentricity.
The case yields a circle and is included as a special type of ellipse.
The equation can be viewed in a different way :
is called the circular directrix (related to focus of the ellipse. This property should not be confused with the definition of an ellipse using a directrix line [|below].
Using Dandelin spheres, one can prove that any 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.

In Cartesian coordinates

Standard equation

The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the x-axis is the major axis, and:
For an arbitrary point the distance to the focus is and to the other focus. Hence the point is on the ellipse whenever:
Removing the radicals by suitable squarings and using produces the standard equation of the ellipse:
or, solved for y:
The width and height parameters are called the semi-major and semi-minor axes. The top and bottom points are the co-vertices. The distances from a point on the ellipse to the left and right foci are and.
It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.

Parameters

Principal axes

Throughout this article, the semi-major and semi-minor axes are denoted and, respectively, i.e.
In principle, the canonical ellipse equation may have . This form can be converted to the standard form by transposing the variable names and and the parameter names and

Linear eccentricity

This is the distance from the center to a focus:.

Eccentricity

The eccentricity can be expressed as:
assuming An ellipse with equal axes has zero eccentricity, and is a circle.

Semi-latus rectum

The length of the chord through one focus, perpendicular to the major axis, is called the latus rectum. One half of it is the semi-latus rectum. A calculation shows:
The semi-latus rectum is equal to the radius of curvature at the vertices.

Tangent

An arbitrary line intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line, tangent and secant. Through any point of an ellipse there is a unique tangent. The tangent at a point of the ellipse has the coordinate equation:
A vector parametric equation of the tangent is:
Proof:
Let be a point on an ellipse and be the equation of any line containing. Inserting the line's equation into the ellipse equation and respecting yields:
There are then cases:

Then line and the ellipse have only point in common, and is a tangent. The tangent direction has perpendicular vector, so the tangent line has equation for some. Because is on the tangent and the ellipse, one obtains.
Then line has a second point in common with the ellipse, and is a secant.

Using one finds that is a tangent vector at point, which proves the vector equation.
If and are two points of the ellipse such that, then the points lie on two conjugate diameters.

Shifted ellipse

If the standard ellipse is shifted to have center, its equation is
The axes are still parallel to the x- and y-axes.

General ellipse

In analytic geometry, the ellipse is defined as a quadric: the set of points of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation
provided
To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant
Then the ellipse is a non-degenerate real ellipse if and only if C∆ < 0. If C∆ > 0, we have an imaginary ellipse, and if ∆ = 0, we have a point ellipse.
The general equation's coefficients can be obtained from known semi-major axis, semi-minor axis, center coordinates, and rotation angle using the formulae:
These expressions can be derived from the canonical equation
by a Euclidean transformation of the coordinates :
Conversely, the canonical form parameters can be obtained from the general-form coefficients by the equations:
where is the 2-argument arctangent function.

Parametric representation

Standard parametric representation

Using trigonometric functions, a parametric representation of the standard ellipse is:
The parameter t is not the angle of with the x-axis, but has a geometric meaning due to Philippe de La Hire.

Rational representation

With the substitution and trigonometric formulae one obtains
and the rational parametric equation of an ellipse
which covers any point of the ellipse except the left vertex.
For this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing The left vertex is the limit
Alternately, if the parameter is considered to be a point on the real projective line, then the corresponding rational parametrization is
Then
Rational representations of conic sections are commonly used in computer-aided design.

Tangent slope as parameter

A parametric representation, which uses the slope of the tangent at a point of the ellipse
can be obtained from the derivative of the standard representation :
With help of trigonometric formulae one obtains:
Replacing and of the standard representation yields:
Here is the slope of the tangent at the corresponding ellipse point, is the upper and the lower half of the ellipse. The vertices, having vertical tangents, are not covered by the representation.
The equation of the tangent at point has the form. The still unknown can be determined by inserting the coordinates of the corresponding ellipse point :
This description of the tangents of an ellipse is an essential tool for the determination of the orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.

General ellipse

Another definition of an ellipse uses affine transformations:
;Parametric representation
An affine transformation of the Euclidean plane has the form, where is a regular matrix and is an arbitrary vector. If are the column vectors of the matrix, the unit circle,, is mapped onto the ellipse:
Here is the center and are the directions of two conjugate diameters, in general not perpendicular.
;Vertices
The four vertices of the ellipse are, for a parameter defined by:
This is derived as follows. The tangent vector at point is:
At a vertex parameter, the tangent is perpendicular to the major/minor axes, so:
Expanding and applying the identities gives the equation for
;Area
From Apollonios theorem one obtains:
The area of an ellipse is
;Semiaxes
With the abbreviations
the statements of Apollonios's theorem can be written as:
Solving this nonlinear system for yields the semiaxes:
;Implicit representation
Solving the parametric representation for by Cramer's rule and using, one obtains the implicit representation
Conversely: If the equation
of an ellipse centered at the origin is given, then the two vectors
point to two conjugate points and the tools developed above are applicable.
Example: For the ellipse with equation the vectors are
;Rotated standard ellipse
For one obtains a parametric representation of the standard ellipse rotated by angle :
;Ellipse in space
The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows to be vectors in space.