Circle

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Definition

As a subset of the Euclidean plane

A circle with center $O$ and radius $r>0$ is defined as the set of all points $P$ in the Euclidean plane at a distance of $r$ from $O$.

The unit circle is the circle whose center is at the origin and radius is $1$, it is defined as the following subset of the Euclidean plane:

$\{(x,y)\mid x^2+y^2=1\}$

Under the identification of the Euclidean plane with the complex numbers, this can also be described as the set of complex numbers whose modulus is $1$.

Note that all circles are equivalent up to similarity transformations of the Euclidean plane.

Equivalent spaces

Space	How strongly is it equivalent to the circle?
Ellipse in ${\mathbb{R}}^2$	equivalent up to an affine transformation
Simple closed convex curve of ${\mathbb{R}}^2$	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$ arising from a straight line homotopy
Simple closed curve in ${\mathbb{R}}^2$	equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$
Compact differential 1-manifold	Diffeomorphic
Compact 1-manifold	Homeomorphic