Delaunay Triangulation: Algorithm and Dynamic Implementation

Delaunay triangulation represents a cornerstone algorithm for spatial partitioning, extensively applied across domains including computer graphics, finite element analysis, and computational physics. This document elaborates on Dynamic Delaunay triangulation – an advanced variant facilitating real-time modifications to input point clouds through structured matrix-based point storage. Implementation typically involves maintaining a dynamic data structure (such as a doubly-connected edge list) and leveraging flip algorithms for local retriangulation when points are inserted or deleted. The algorithm's robustness stems from its mathematical properties, particularly the empty circumcircle criterion: no triangulation vertex resides inside the circumcircle of any triangle. This guarantees maximized minimum angles, producing numerically stable mesh configurations. Key functions in practice include point location queries via walk algorithms and incremental insertion with Bowyer-Watson methodology, making it indispensable for applications requiring adaptive mesh refinement.