closegeometry

Closegeometry is a term used to describe the study of proximity relations and closeness measures between geometric objects within a metric framework. In practice, it concerns how near two shapes, points, or structures are to one another, and how small perturbations or measurement errors influence those relationships. The central aim is to quantify closeness with distance functions and to formalize neighborhoods and tolerances that reflect usable similarity in computation and analysis.

Formally, closegeometry relies on distance notions such as the Euclidean distance, other Lp metrics, and the Hausdorff distance between sets. Objects can be points, curves, surfaces, or higher-dimensional shapes, and closeness can be defined using point-to-point, set-to-set, or feature-based measurements. Thresholds generate epsilon-neighborhoods and enable the construction of proximity relations and graphs.

Key problems and methods in closegeometry include the closest pair problem, nearest-neighbor search, range searching, and the construction of proximity graphs such as the relative neighborhood graph or Gabriel graph. Practitioners seek efficient algorithms that scale to large data sets and to high dimensions, while remaining robust to noise and perturbations through stable representations and descriptors.

Applications span computer graphics, collision detection, robotics and motion planning, geographic information systems, and pattern recognition. In mesh processing and computer-aided design, closegeometry informs noise filtering, shape matching, and feature alignment by emphasizing small but significant geometric differences rather than exact equality.

Closegeometry is not an established formal subdiscipline with universal axioms; rather, it is a descriptive label for a family of problems linked to proximity, stability, and metric analysis within geometry and computational geometry. Its scope overlaps with metric geometry, topology-inspired shape analysis, and data-driven geometric inference.

Related topics include metric geometry, computational geometry, distance measures, Hausdorff distance, and shape matching.