sfere

Sfere, in geometry, are the set of all points in three-dimensional Euclidean space that are at a fixed distance, called the radius, from a central point. The radius is the only parameter needed to determine a sfere; the set forms the boundary of a solid region known as a ball, while the interior is the solid ball. The sfere itself is a two-dimensional surface.

The 2D analogue is the circle, and in higher dimensions the generalization is called the n-sphere. A common way to describe a sfere with center (a,b,c) and radius r is by the equation (x−a)^2+(y−b)^2+(z−c)^2 = r^2. Intersections with planes yield circles, and cross-sections through the center are great circles.

Key properties include that every point on the sfere is at distance r from the center, and that many cross-sections are circles. In higher dimensions, the set of points at distance r from a center in R^{n+1} is called the n-sphere, denoted S^n.

Formulas: the surface area of a sfere with radius r is 4πr^2, and the volume enclosed by the sfere (the solid ball) is 4/3 πr^3. The unit sfere has r = 1.

Applications: sfere are foundational in geometry and appear in physics, computer graphics, geodesy, and various modeling contexts, serving as idealized representations of round objects and celestial bodies.