Ray-Marching Thurston Geometries

Abstract

We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry X, we also consider scenes within quotient manifolds and orbifolds $X/\Gamma$. We adapt the Phong lighting model to non-euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Acknowledgments

We thank Joey Chahine for telling us about a computable means of finding area density. We thank Arnaud Chéritat, Matei Coiculescu, Jason Manning, Saul Schleimer, and Rich Schwartz for enlightening discussions about the Thurston geometries at ICERM.

Correction statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 There are 19 maximal geometries in dimension four [25], and 58 in dimension five [19]. While many of these can be constructed by analogous procedures, some new phenomena also arise.

2 In practice, we allow a march of the distance to the nearest wall plus some small ε: this prevents wasting many steps approaching the boundary to no appreciable theoretical disadvantage: the teleportation scheme returns us to D immediately upon overstep.

3 A commonly cited reference for solving the geodesic flow in Sol is [7]. However, the authors do not conduct the computation to the final stage – see their Theorem 4.1(1). Moreover, the formulas given in Theorem 4.1(2) have some errors.

Additional information

Funding

This material is based in part upon work supported by the National Science Foundation under Grant No. DMS-1439786 and the Alfred P. Sloan Foundation award G-2019-11406, while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Illustrating Mathematics program. The first author acknowledges support from the Centre Henri Lebesgue ANR-11-LABX-0020-01 and the Agence Nationale de la Recherche under Grant Dagger ANR-16-CE40-0006-01. The second author was supported in part by National Science Foundation grant DMR-1847172 and a Cottrell Scholars Award from the Research Corporation for Science Advancement. The third author was supported in part by National Science Foundation grant DMS-1708239.