nearDelaunay

NearDelaunay is a term used in computational geometry to describe a class of triangulations or proximity graphs that approximate the Delaunay triangulation while relaxing the strict empty-circumcircle condition. In a nearDelaunay triangulation, edges are chosen so that each triangle is nearly Delaunay: its circumcircle is allowed to contain a limited number of other input points, or the circle is allowed to intersect the point set within a controlled tolerance. The exact definition varies by source, but the core idea is to preserve much of the favorable geometry of Delaunay triangles—such as avoiding extremely skinny triangles and maintaining relatively uniform edge lengths—while improving robustness on degenerate or noisy data.

NearDelaunay graphs may be generated by perturbing the input points slightly, by relaxing the empty-circle criterion with a parameter epsilon, or by permitting additional edges beyond the Delaunay triangulation under a quality constraint. They therefore form supersets of Delaunay triangulations and can be tuned via a tolerance parameter to trade strict optimality for stability and mesh quality.

Applications include mesh generation for finite element methods, geographic information systems, and computer graphics, where exact Delaunay triangulations can be brittle in practice due to noise or degeneracies. Algorithms for constructing nearDelaunay structures often adapt standard Delaunay tools—incremental insertion, edge flipping, or constrained triangulations—with added checks or perturbations to enforce the near-Delaunay condition.

Relation to other graphs: the Delaunay triangulation is a special case of nearDelaunay when the tolerance is zero. NearDelaunay concepts are commonly compared with Gabriel and relative neighborhood graphs to study edge inclusion and geometric properties.