DistancetoA

DistancetoA is a term used in mathematics, computer science, and related fields to describe the distance from a point p in a space to a fixed reference point A. It is typically denoted dist(p, A) or d(p, A). When the underlying space is a metric space with metric d, distToA(p) is defined as d(p, A). This creates a scalar field over the space that measures how far each point lies from A.

In two or more dimensions, distToA depends on the chosen metric. For Euclidean space with p = (x, y) and A = (a, b), the Euclidean distance is distToA(p) = sqrt((x − a)^2 + (y − b)^2). Other common metrics yield distToA(p) = |x − a| + |y − b| for the Manhattan (L1) distance, or distToA(p) = max(|x − a|, |y − b|) for the Chebyshev (L∞) distance. Regardless of the metric, distToA is a real-valued function on the space indicating proximity to A.

Computationally, distToA is used in algorithms for nearest-neighbor search, clustering, and spatial reasoning. Efficient evaluation often relies on spatial indices such as kd-trees, ball trees, or grid-based methods to reduce the number of distance computations when working with large datasets.

Applications of distToA include path planning, where distance to a goal matters, Voronoi diagram construction with respect to a fixed site, and in machine learning as a simple radial feature or as part of prototype-based methods. Related concepts include distance functions, metrics, and distance to a centroid or prototype.