Circle

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

A circle is a 1-dimensional surface produced by finding the set of all points that are an equal distance from another point in 2-dimensional space. Because it is curved, it is often represented embedded in the plane. A circle is the shape of the exterior of a disc. It is the 1-dimensional hypersphere.

It can also be considered to be a torus with the base as a point, or alternately as a line segment curved such that the two ends connect.

A circle is an example of a conic section, and is created when a plane parallel to the base cuts the cone.

Structure and Sections

Hypervolumes

• vertex count = $0$
• edge length = $\pi l$

Subfacets

• 1 circle (1D)

Embeddings

ℝ²

A circle can be defined parametrically using the parameter $t$ by

$\begin{align} x(t) = r \cos t \\ y(t) = r \sin t \\ \end{align}$

Where r is a constant defining the radius of the circle. Squaring both of these and adding them together gives the cartesian form of the circle with radius r,

$x^2 + y^2 - r^2 = 0$

ℂ¹

A circle can also be embedded in a complex coordinate space using the parameter $t$ by

$\begin{align} z(t) = r e^{it} \end{align}$

See Also

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	...	Aleph null
Hyperbolic space $\mathbb H^{n}$	—	—	—	Hyperbolic plane $\mathbb H^{2}$	Hyperbolic realm $\mathbb H^{3}$	Hyperbolic flune $\mathbb H^{4}$	Hyperbolic pentrealm $\mathbb H^{5}$	Hyperbolic hexealm $\mathbb H^{6}$	Hyperbolic heptealm $\mathbb H^{7}$	Hyperbolic octealm $\mathbb H^{8}$	Hyperbolic ennealm $\mathbb H^{9}$	Hyperbolic decealm $\mathbb H^{10}$	...	Hyperbolic omegealm $\mathbb H^{\aleph_0}$
Euclidean space $\R^n$	Null polytope $\emptyset$	Point $\mathbb R^{0}$	Euclidean line $\mathbb R^{1}$	Euclidean plane $\mathbb R^{2}$	Euclidean realm $\mathbb R^{3}$	Euclidean flune $\mathbb R^{4}$	Euclidean pentrealm $\mathbb R^{5}$	Euclidean hexealm $\mathbb R^{6}$	Euclidean heptealm $\mathbb R^{7}$	Euclidean octealm $\mathbb R^{8}$	Euclidean ennealm $\mathbb R^{9}$	Euclidean decealm $\mathbb R^{10}$	...	Euclidean omegealm $\mathbb R^{\aleph_0}$
Hypersphere $\mathbb S^{n}$		Point pair $\mathbb S^{0}$	Circle $\mathbb S^{1}$	Sphere $\mathbb S^{2}$	Glome $\mathbb S^{3}$	Tetrasphere $\mathbb S^{4}$	Pentasphere $\mathbb S^{5}$	Hexasphere $\mathbb S^{6}$	Heptasphere $\mathbb S^{7}$	Octasphere $\mathbb S^{8}$	Enneasphere $\mathbb S^{9}$	Dekasphere $\mathbb S^{10}$	...	Omegasphere $\mathbb S^{\aleph_0}$