closestPoint

Closestpoint, often written as closest point or closestPoint in programming libraries, denotes the point in a given set that is nearest to a specified external point. In mathematics, for a set S in Euclidean space and a query point q, the closest point p* is defined as p* = argmin_{p in S} ||q − p||, and the distance to the set is d(q, S) = min_{p in S} ||q − p||. If q belongs to S, the distance is zero and the closest point is q itself. If S is closed and nonempty, the minimum exists; if S is convex, the closest point is unique.

Properties of the closest point depend on the geometry of S. For smooth, boundary points, the vector from q to p* is normally aligned with the surface, meaning q − p* is orthogonal to the boundary at p*. For convex sets, projection mappings are well-behaved: they are non-expansive and define a continuous, single-valued projection operator.

Computationally, closestpoint problems appear as nearest-neighbor queries and projection tasks. With discrete data, efficient methods include kd-trees, ball trees, and other spatial indexes to locate the nearest data point. For continuous surfaces or volumes, projection may require solving optimization problems or using specialized formulas for simple shapes (planes, spheres, cylinders) or iterative methods for complex surfaces. In computer graphics and robotics, closestpoint queries underpin collision detection, distance fields, mesh processing, and motion planning.

See also: nearest neighbor search, projection, distance function.