Polygonal Sequence-driven Triangulation Validator: An Incremental Approach to 2D Triangulation Verification

TL;DR

The paper tackles the problem of verifying 2D Delaunay triangulations produced by floating-point computations, where rounding errors can yield incorrect triangulations or violate the Delaunay property. It introduces PSTV, an incremental, polygon-sequence-based validator that builds a triangulation frontier $\{P_k,B_k\}$ and uses fast data structures, including an edge map and interval trees, to efficiently locate adjacent triangles and verify adjacency, orientation, and incircle predicates with floating-point filters. Robustness is achieved via Ozaki’s orientation filter and an extended Shewchuk incircle filter that handles underflow, with rare exact arithmetic via GMP when necessary; the approach avoids interval arithmetic to maintain performance. The method also includes a post hoc correction step to enforce the Delaunay property through edge flipping (respecting constrained edges) and demonstrates its practical viability through numerical experiments and a web application, offering a posteriori guarantees useful for FEM discretizations and computer-assisted proofs. The work provides a concrete pathway to reliable triangulation verification independent of the generating software, with empirical efficiency and broad applicability in computational geometry and numerical analysis. $P_h$ denotes the mesh and $C_h$ the interpolation error constant relevant to FEM, highlighting the practical impact of guaranteeing triangulation quality on numerical solutions.$

Abstract

Two-dimensional Delaunay triangulation is a fundamental aspect of computational geometry. This paper presents a novel algorithm that is specifically designed to ensure the correctness of 2D Delaunay triangulation, namely the Polygonal Sequence-driven Triangulation Validator (PSTV). Our research highlights the paramount importance of proper triangulation and the often overlooked, yet profound, impact of rounding errors in numerical computations on the precision of triangulation. The primary objective of the PSTV algorithm is to identify these computational errors and ensure the accuracy of the triangulation output. In addition to validating the correctness of triangulation, this study underscores the significance of the Delaunay property for the quality of finite element methods. Effective strategies are proposed to verify this property for a triangulation and correct it when necessary. While acknowledging the difficulty of rectifying complex triangulation errors such as overlapping triangles, these strategies provide valuable insights on identifying the locations of these errors and remedying them. The unique feature of the PSTV algorithm lies in its adoption of floating-point filters in place of interval arithmetic, striking an effective balance between computational efficiency and precision. This research sets a vital precedent for error reduction and precision enhancement in computational geometry.

Figures (18)

• Figure 1: Triangulation (left) is obtained from the built-in 'delaunay' function in SciPy. The highlighted part (right) shows overlapping triangles. The failure of the triangulation process involves overlapping, which results in shapes that are not even triangles. In relation to $\triangle{BCD}$, $\triangle{ABC}$ and $\triangle{ACD}$ overlap. The versions of SciPy and Python used for this triangulation are 1.11.0 and 3.9, respectively.
• Figure 2: Finite element solution of the Poisson equation solved on the incorrect triangulation depicted in Fig. \ref{['fig:wrong-tri-1']}. The corresponding matrix equations were solved and the visualization was produced using MATLAB R2022b.
• Figure 3: Triangulation obtained from the 'buildmesh' function in FreeFEM++ v4.9. The areas highlighted by the circles do not satisfy the local Delaunay property.
• Figure 4: Example of orientation test (point $p_c$ is to the left of the oriented line segment $\overrightarrow{p_a p_b}$).
• Figure 5: Example of incircle test (point $p_d$ lies inside the circumcircle of triangle $p_a p_b p_c$).

Theorems & Definitions (1)

• Definition 1