Work

Sums of Wedges: Conforming Weighted Delaunay Triangulations are Polynomial in Fixed Dimension

Year: 2025

Type: article

Abstract: We show how the problem of creating a triangulation in d -dimensional space that conforms to constraints given as sub-simplices can be turned into the problem of computing the lower hull of a sum of wedge functions. This sum can be interpreted as a Weighted Delaunay Triangulations, necessarily containing the constraints as unions of its elements. Intersections of wedges lead to Steiner points. As the number of such intersections is polynomial in the number of wedges, and the number of wedges per element is typically 1 (at most d ), this proves that the complexity of the output is polynomial. Moreover, we show that the majority of wedge intersections is unnecessary for a conforming triangulation and further heuristically reduce the number of Steiner points. Using appropriate data structures, the function can be evaluated in quasi-linear time, leading to an output-sensitive algorithm.

Source: ACM Transactions on Graphics

DOI: 10.1145/3763368

Authors: Dimitrios Bogiokas, Ugo Finnendahl, Thorsten Seidelmann, Marc Alexa

Institution: Technische Universität Berlin

Language: English

Topic: Computational Geometry and Mesh Generation

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Funders: Deutsche Forschungsgemeinschaft, Berlin Mathematics Research Center MATH+, European Research Council

Awards: G3517083867, G8964763783