Spheresuch

Spheresuch is a term used in geometry to describe the family of spheres that pass through all points of a given finite subset X of Euclidean space. For a set X in R^3, spheresuch(X) denotes the collection of all spheres with center c and radius r such that the distance from c to every point p in X equals r. The concept is often discussed in the context of circumscribed spheres and three-dimensional reconstruction.

Basic properties and special cases. If X consists of four non-coplanar points that lie on a common sphere, spheresuch(X) contains exactly one sphere. If the points do not lie on a single sphere, spheresuch(X) is empty. If X lies in a plane and all points lie on a circle, there are infinitely many spheres whose centers lie on the line perpendicular to the plane through the circle’s center; each such sphere has the same radius equal to the circle’s radius. More generally, the size of spheresuch(X) reflects the degree to which the points in X are co-spherical.

Computation and construction. Determining spheresuch(X) reduces to solving a set of equations for the center coordinates and radius. In practice, numerical methods or algebraic approaches are used to check feasibility and, when possible, compute the unique sphere or the family of spheres.

Applications. The concept is used in three-dimensional modeling, computer vision, sensor calibration, and geometric design. It also serves as a theoretical tool in studies of sphericity and circumscribed structures.

See also. Circumsphere, Sphere through points, Spherical hull.