Constrained Delaunay Tetrahedrization: A Robust and Practical Approach

TL;DR

This paper introduces a robust, parameter-free algorithm for constrained Delaunay tetrahedrization that achieves a 100% success rate on a large dataset, simplifying implementation and ensuring geometric conformity.

Contribution

It presents a floating-point based, robust CDT algorithm using indirect geometric predicates, addressing theoretical gaps and eliminating ad-hoc tolerances in geometric computations.

Findings

Achieved 100% success on Thingi10k dataset models.

Simplified implementation by removing ad-hoc tolerances.

Produced conforming, valid tetrahedral meshes with minimal Steiner points.

Abstract

We present a numerically robust algorithm for computing the constrained Delaunay tetrahedrization (CDT) of a piecewise-linear complex, which has a 100% success rate on the 4408 valid models in the Thingi10k dataset. We build on the underlying theory of the well-known TetGen software, but use a floating-point implementation based on indirect geometric predicates to implicitly represent Steiner points: this new approach dramatically simplifies the implementation, removing the need for ad-hoc tolerances in geometric operations. Our approach leads to a robust and parameter-free implementation, with an empirically manageable number of added Steiner points. Furthermore, our algorithm addresses a major gap in TetGen's theory which may lead to algorithmic failure on valid models, even when assuming perfect precision in the calculations. Our output tetrahedrization conforms with the input geometry without approximations. We can further round our output to floating-point coordinates for downstream applications, which almost always results in valid floating-point meshes unless the input triangulation is very close to being degenerate.