Euclidean circle
= Euclidean circle
{c}
{title2=$\{x\in\mathbb R^2:\lVert x-c\rVert=r\},\quad r>0$}
A \b[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 $c$:
$$
C(c,r)=\{x\in\mathbb R^2:\lVert x-c\rVert=r\}.
$$
It is the boundary curve, while the <closed disk> includes all points with $\lVert x-c\rVert\leq r$. 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.<Euclid's Elements by Euclid>[BookI, Definitions15–18, Joyce transcription and guide distinguishing Euclid's figure from the modern boundary curve]{ref}