This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

Circle

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In Euclidean geometry:

a circle is a set of points equidistant from a given point.

On the $xy$-plane, the unit circle at $0$ is given by $$x^2+y^2=1.$$

More generally, in the Cartesian plane: $$(x-a)^2+(y-b)^2=R^2$$ is the circle of radius $r$ centered in point $(a,b)$.

In a normed space with norm $||.||$: $$\{u:||u-a||=R\} $$ is the circle of radius $R$ centered in point $a$.

In metric space $(M,d)$: $$\{u:d(u,a)=R\} $$ is the circle of radius $R$ centered in point $a$.

As a parametric curve on the plane: $$x=\cos t, y=\sin t.$$ Its curvature is 1.

In polar coordinates: $$r=R$$.

The topological circle ${\bf S}^1$ is compact, connected, and its Betti numbers are 1,1,0,...