Efficient Embeddings in Exact Arithmetic

TU Berlin, Computer Graphics Group

teaser

Abstract

We provide a set of tools for generating planar embeddings of triangulated topological spheres. The algorithms make use of Schnyder labelings and realizers. A new representation of the realizer based on dual trees leads to a simple linear time algorithm mapping from weights per triangle to barycentric coordinates and, more importantly, also in the reverse direction. The algorithms can be implemented so that all coefficients involved are $1$ or $-1$. This enables integer computation, making all computations exact. Being a Schnyder realizer, mapping from positive triangle weights guarantees that the barycentric coordinates form an embedding. The reverse direction enables an algorithm for fixing flipped triangles in planar realizations, by mapping from coordinates to weights and adjusting the weights (without forcing them to be positive). In a range of experiments, we demonstrate that all algorithms are orders of magnitude faster than existing robust approaches.

Resources

$\mathrm{B{\footnotesize IB}}{ \TeX}$

@article{Finnendahl_and_Worchel:2025:DiffAcousticPT,
author = {Finnendahl, Ugo and Worchel, Markus and Jüterbock, Tobias and Wujecki, Daniel and Brinkmann, Fabian and Weinzierl, Stefan and Alexa, Marc},
title = {Differentiable Geometric Acoustic Path Tracing using Time-Resolved Path Replay Backpropagation},
year = {2025},
issue_date = {August 2025},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {44},
number = {4},
issn = {},
url = {https://doi.org/10.1145/3730900},
doi = {10.1145/3730900},
abstract = {},
journal = {ACM Trans. Graph.},
month = {aug},
articleno = {},
numpages = {},
keywords = {differentiable rendering, geometrical acoustics, physically-based simulation, acoustic optimization}
}