Visualization of Nil, Sol, and SL2(R)˜ geometries

Adapting Game Engines to Curved Spaces

“…If ray marching or its improved versions like, e.g. sphere tracing, is used instead, non-isotropic Thurston geometries [6,20] or even geometries with varying curvature can also be rendered [8]. Visualizing special and general relativity or black holes are good examples for these approaches [30,31].…”

How to see the eight Thurston geometries

“…This section provides an introduction to Riemannian ray tracing [40,42], a computer graphics technique that allows rendering of images inside a Riemannian manifold (M, g). We will use this concept in Chapter 6 to visualize scenes embedded in 3-manifolds modeled by Thurston's geometries.…”

“…We launch a ray γ towards each vector v ∈ V p . If γ hits a visible object (red triangle) in γ(|v|) = q we define an RGB color for the corresponding point in the near plane of V p by considering the direct and indirect illumination (image from [42]). On the right, the hit point q is connected with the light source l through a geodesic.…”

“…Specifically, the scene objects are set in 3-manifold spaces. Recent works [6,40,42,61] avoid modeling perspective views by using a generalization of the ray tracing algorithm: a color is given to each point and tangent direction by tracing a ray and finding its intersections with the scene objects. Recall from physics that light travels along the rays -the paths that locally minimize length.…”

“…Recently, a framework from [10,42,39,61,43] implemented on top of Nvidia RTX was introduced for real-time immersive and interactive visualization of spaces modeled by Thurston's geometries and other nonhomogeneous spaces. Other important attempts to visualize and explore non-Euclidean geometries are presented in [7,8,9,30].…”