sphère

A sphere is the set of all points in three-dimensional Euclidean space that are at a fixed distance, called the radius, from a given point, the center. This definition refers to the hollow surface; the solid region inside is called a ball (boule).

In Cartesian coordinates, a sphere with center (a, b, c) and radius r is described by (x − a)^2 + (y − b)^2 + (z − c)^2 = r^2. The surface is two-dimensional, while the solid ball consists of all points with (x − a)^2 + (y − b)^2 + (z − c)^2 ≤ r^2. In higher dimensions, an n-dimensional sphere, denoted S^n, is the set of points at distance r from a fixed center in R^{n+1}.

Key properties include uniform curvature and high symmetry: all surface points are equivalent under rotations about the center. Geodesics on the sphere are great circles, and the spherical distance is measured along the surface.

Etymology: the term comes from Greek sphaira, meaning globe or ball. In French usage, la sphère usually refers to the hollow surface, while la boule denotes the solid region.

Applications and appearances: spheres model celestial bodies, serve as standard shapes in computer graphics and geometry, appear in spherical coordinates and in sphere packing problems, and are used in optics and physical design to approximate round objects. The Earth is nearly spherical, but slightly oblate.