Withingeom

Withingeom is a conceptual framework for reasoning about geometric relations that emphasizes the internal properties of shapes, such as containment and interior–boundary interactions. The name is a blending of "within" and "geometry," and it appears mainly in instructional contexts and experimental libraries rather than as a formal standard.

Core ideas include the explicit treatment of within relationships—whether a point lies inside a shape, whether one region is entirely contained in another, and how interiors change under transformations. Operations commonly discussed in withingeom-style discussions include containment tests, boundary evaluations, and the composition of regions via union, intersection, and difference, with a focus on the interior and boundary as first-class features.

Shapes can be represented in withingeom by polygons, meshes, implicit surfaces, or signed distance fields. The framework supports typical algorithms such as point-in-polygon tests and winding-number methods, as well as generalized interior tests for curved or topologically complex regions. Numerical robustness and degeneracies are noted as important considerations.

Applications of withingeom appear in education, visualization tools, and experimental CAD or GIS research, where understanding how interiors relate to each other is valuable. It provides a language for describing inside/outside predicates independent of a specific representation.

Status notes: withingeom is not an official mathematical or engineering standard. It is used informally to discuss containment concepts and as a pedagogical device. There are no universal specifications, and implementations vary across projects.