Ellipse

Ellipse
Dimensionality	1
Subfacets
Edges	1 ellipse
Hypervolumes
Vertex Count	'"`UNIQ--postMath-00000001-QINU`"'
Edge Length	'"`UNIQ--postMath-00000002-QINU`"'

An ellipse is a 1-dimensional surface produced by finding the set of points around a point pair of two focal points, where the sum of the two distances from the points stay constant.

The longest distance from the edge to the centre is the semi-major axis $a$ and the shortest distance is the semi-minor axis $b$.

As an ellipse is curved, it is often represented embedded in a plane.

A circle can be considered an ellipse where the focal points are one and the same point.

An ellipse is often mistaken for an oval, a closed curve that only loosely resembles an egg. The main difference between the two is that an ellipse requires the mathematical definition, while an oval doesn't. Therefore, all ellipses are ovals, but not all ovals are ellipses.

An ellipse is an example of a conic section, and is created when the dividing plane cuts the cone at an angle, while the plane doesn't intersect with the base.

Edge Length

The perimeter of an ellipse can not be written in a closed form in terms of elementary functions. However, it can be written simply using elliptic integrals, the definition of which was partially motivated by the expression for the perimeter of an ellipse

$4a E\left(\sqrt{1-\frac{a^2}{b^2}}\right)$

Where $E$ is the complete elliptic integral of the second kind.

This can be expressed as an infinite series in various ways. One such infinite series is

$2a\pi(1-\sum_{i=1}^{\infty}(\frac{(2i)!^2}{(2^i \cdot i!)^4} \cdot {(\frac {(\frac {\sqrt {a^2 - b^2}}{a})^{2i}}{2i - 1})}))$

Structure and Sections

Hypervolumes

• vertex count = $0$
• edge length = $4a E\left(\sqrt{1-\frac{a^2}{b^2}}\right)$

Subfacets

• 1 ellipse (1D)