pointwhere

Pointwhere is a conceptual term used in geometry, spatial analysis, and data science to describe the locus of points that satisfy a specified condition relative to a reference point or reference field. The idea generalizes traditional notions such as level sets and loci by combining spatial relationships with attribute constraints.

Definition. Consider a metric space X with a real-valued function f: X → R, a reference point p ∈ X, and a predicate P that may depend on a point x ∈ X, its attribute f(x), and possibly p. The pointwhere set associated with these inputs is W(p, f, P) = { x ∈ X | P(x, f(x), p) holds }. If P(x, f(x), p) expresses equality f(x) = f(p), then W is a level set of f at the value f(p). If P additionally imposes a neighborhood constraint such as dist(x, p) ≤ r, then W becomes the portion of that level set within the radius r around p.

Key properties. Pointwhere sets accommodate multiple conditions, enabling intersections of level-set behavior with spatial proximity or other attributes. They can form curves, surfaces, or more complex subspaces, depending on the space X and the predicate P. When P is simple and f is continuous, W often inherits regularity from f, such as being a manifold or a finite union of manifolds in regions where f behaves smoothly.

Relation to other concepts. Pointwhere relates closely to level sets, contours, and loci, but emphasizes the role of a reference point and composite predicates. It also connects to isosurfaces in higher dimensions and to neighborhood-based filtering in data analysis.

Applications. Used in pattern recognition, boundary detection, geospatial querying, robotics, and visualization to describe regions defined by specific attribute values around a focal point or within a targeted neighborhood. See also level set, locus, contour, and isosurface.