Ray-marching Thurston Geometries

Technical paper on writing interactive computer graphics in curved space.

AuthorsRemi Coulon, Sabetta Matsumoto, Henry Segerman, Steve Trettel
JournalExperimental Mathematics
CategoriesTopology, Geometry, Computer Graphics


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.

Read on arXiv