 Euclidean circle

A Euclidean circle with center $c\in {\mathbb{R}}^2$ and radius $r>0$ is the set of points in the Euclidean plane at Euclidean distance exactly $r$ from $c$:It is the boundary curve, while the closed disk includes all points with $\Vert x-c\Vert \le r$. The center belongs to that closed disk but not to the Euclidean circle.

For example, center $c=(2,-1)$ and radius3 give $(x-2{)}^2+(y+1{)}^2=9$. The point $(5,-1)$ lies on the Euclidean circle; $(2,1)$ lies inside the closed disk because its distance from the center is2. Such an equation distinguishes a distance constraint from a region-filling inequality.

Every Euclidean circle is a circle in the topological sense. The map $u\mapsto c+ru$ from the unit circle $S^1$ is a homeomorphism, with inverse $x\mapsto (x-c)/r$. Topology forgets the particular center and radius, whereas the Euclidean distance retains them. The assumption $r>0$ matters: radius zero gives a point, which is not a circle.

The historical terminology differs slightly. Euclid's definition describes the plane figure contained by a circumference; the modern set above corresponds to that boundary. His equal-distance condition and distinguished center appear in BookI, Definitions15–18.^[ref]