EdgeSammelpunkte

EdgeSammelpunkte, translated as edge-accumulation points, is a concept used in computational geometry and graph theory to describe the limiting behavior of edge endpoints in a sequence of geometric graphs or polygonal meshes. In this framework, one studies how the locations of edge endpoints distribute as the graph or mesh is refined or as samples are taken more densely.

Definition. Let G_k = (V_k, E_k) be a sequence of graphs embedded in R^n. A point x in R^n is called an EdgeSammelpunkte if every neighborhood of x contains endpoints of edges from infinitely many graphs in the sequence. Equivalently, x belongs to the accumulation (or cluster) set of edge endpoints across the sequence.

Properties. The EdgeSammelpunkte set is typically closed and can lie on domain boundaries or appear along internal curves, depending on the refinement strategy and edge distribution. It can be simple, containing a finite number of points, or more complex, potentially forming continuous or fractal-like sets under certain refinements. The concept helps distinguish regions where edge endpoints concentrate as refinement proceeds.

Computation. In practice, EdgeSammelpunkte are approximated by gathering edge endpoints from the first several graphs and computing their limit points via clustering or density estimation. The observed dense regions provide an estimate of the accumulation set, and the closure of these regions serves as an approximation to the theoretical EdgeSammelpunkte.

Applications. The notion is used in mesh refinement analysis, convergence studies of geometric algorithms, mesh quality control, and path-planning models where edge distributions reflect potential bottlenecks or critical regions.

See also. accumulation point, limit point, mesh refinement, geometric graph, edge endpoint.