Bounding sphere

Some instances of the smallest bounding circle, the case of the bounding sphere in 2 dimensions.

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.[1]

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 a sphere is unique: If there are two of them, then the objects in question lie 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".

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.

There are exact and approximate algorithms for the solving bounding sphere problem.

In 1991, Emo Welzl proposed a much simpler randomized algorithm based in the extension of a randomized linear programming algorithm by Raimund Seidel. It runs in expected linear time. The paper provides experimental results demonstrating its practicality in higher dimensions.[3]

The open-source Computational Geometry Algorithms Library (CGAL) contains an implementation of this algorithm.[4]

In 1990, Jack Ritter proposed a simple algorithm to find a non-minimal bounding sphere.[5] It is widely used in various applications for its simplicity. The algorithm works in this way:

Fischer et al. (2003) proposed an exact solver, though the algorithm does not have a polynomial running time in the worst case.[8] The algorithm is purely combinatorial and implements a pivoting scheme similar to the simplex method for linear programming, used earlier in some heuristics. It starts with a large sphere that covers all points and gradually shrinks it until it cannot be shrunk further. The algorithm features correct termination rules in cases of degeneracies, overlooked by prior authors; and efficient handling of partial solutions, which produces a major speed-up. The authors verified that the algorithm is efficient in practice in low and moderately low (up to 10,000) dimensions and claim it does not exhibit numerical stability problems in its floating-point operations.[8] A C++ implementation of the algorithm is available as an open-source project.[9]