Ray-Marching Thurston Geometries. Issue 4 (2nd October 2022)

• Record Type: Journal Article
• Authors: Coulon, Rémi; Matsumoto, Elisabetta A.; Segerman, Henry; Trettel, Steve J.
• Is Part Of: Experimental mathematics. Volume 31:Issue 4(2022)
• Page Start: 1197
• Page End: 1277
• Publication Date: 2022-10-02
• Subjects: Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2022.2030262
• Languages: English
• ISSNs: 1058-6458

Abstract

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 / Γ . 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.