Bounding sphere

In mathematics, given a non-empty set of objects of finite extension in n-dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is an n-dimensional solid sphere containing all of these objects.

In the plane the terms bounding or enclosing circle are used.

Used in computer graphics and computational geometry, a bounding sphere is a special type of bounding volume. There are several fast and simple bounding sphere construction algorithms with a high practical value in real-time computer graphics applications.

In statistics and operations research, the objects are typically points, and generally the sphere of interest is the minimal bounding sphere, that is, the sphere with minimal radius among all bounding spheres. It may be proven that such sphere is unique. If there are two of them, then the objects in question lies within their intersection. But an intersection of two non-coinciding spheres of equal radius is contained in a sphere of smaller radius. The problem of computing the center of a minimal bounding sphere is also known as the "unweighted Euclidean 1-center problem".

Applications

Clustering

Such spheres are useful in clustering, where groups of similar data points are classified together.

In statistical analysis the scattering of data points within a sphere may be attributed to measurement error or natural (usually thermal) processes, in which case the cluster represents a perturbation of an ideal point. In some circumstances this ideal point may be used as a substitute for the points in the cluster, advantageous in reducing calculation time.

In operations research the clustering of values to an ideal point may also be used to reduce the number of inputs in order to obtain approximate values for NP-hard problems in a reasonable time. The point chosen is not usually the center of the sphere, as this can be biased by outliers, but instead some form of average location such as a least squares point is computed to represent the cluster.

Software for computing the minimal bounding sphere

• Miniball software — C++ software to compute the minimal bounding sphere of a pointset in dimensions up to 30 (distributed under the GPL license)
• Minimal bounding sphere of a set of balls in dimensions up to 30 in CGAL, the Computational Geometry Algorithms Library

See also

• Bounding volume

External links

• Smallest Enclosing Circle Problem – describes several algorithms for enclosing a point set, including Megiddo's linear-time algorithm