MinDistance

MinDistance is a term used in geometry, mathematics, and computer science to denote the smallest distance between two geometric objects or sets. Formally, for two sets A and B in a metric space with distance function d, the min distance is dist(A,B) = inf{ d(a,b) : a in A, b in B }. If A and B share a point, the minimum distance is 0. In Euclidean space, the minimum distance can be realized by a pair (a*, b*) of points when A and B are closed and one is compact; otherwise the infimum may not be attained.

Special cases include point-to-set distance dist(p, A) = inf{ d(p,a) : a in A }.

Directed distance from A to B is often distinguished, defined as inf{ d(a,b) : a in A, b in B }, which may differ from dist(B,A) if the metric is not symmetric.

Computationally, min distance is computed via pairwise distance calculations, segment-to-segment distance for polygonal objects, nearest-neighbor search, or convex optimization in higher dimensions. Algorithms rely on data structures like kd-trees, Voronoi diagrams, or spatial indexing to reduce comparisons.

Applications span collision detection in graphics and robotics, path planning, clustering and pattern recognition, and geometric queries in GIS. In software, a function named minDistance or similarly named computes this quantity for objects such as points, sets, or geometric primitives.

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