Sphere

Remarks

In coordinates, $S(c, r)$ is the set of solutions of $(x_1 - c_1)^2 + \cdots + (x_n - c_n)^2 = r^2$. The ordinary sphere is the case $n = 3$, and the case $n = 2$ is a circle. The unit sphere $S(0, 1)$ in $\R^{n+1}$ is the n-sphere $S^n$. In any metric space $(X, d)$, the same formula defines the sphere $S(c, r)$; Mathlib's EuclideanGeometry.Sphere records a centre and a radius in this way.

Example: the set $x^2 + y^2 + z^2 - 2x = 0$ in $\R^3$ is $S((1, 0, 0), 1)$. Indeed, the equation is $(x - 1)^2 + y^2 + z^2 = 1$.

Non-example: for $r = 0$ the set $\set{x : d(x, c) = 0}$ is the single point $c$. It fails the condition $r > 0$, so it is not a sphere.