Ray-marching Thurston geometries

Rémi Coulon (1) , Elisabetta Matsumoto , Henry Segerman , Steve Trettel

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Domaines: Géométrie métrique [math.MG]; Topologie géométrique [math.GT]; Théorie des groupes [math.GR]

Rémi Coulon, Elisabetta Matsumoto, Henry Segerman, Steve Trettel. Ray-marching Thurston geometries. Experimental Mathematics, 2022, 31 (4), pp.1197-1277. ⟨10.1080/10586458.2022.2030262⟩. ⟨hal-02983618⟩