We present a “real-time” ray tracing algorithm to immersive visualization of manifolds with their geometries locally modeled by Nil, Sol, and SL2(R) geometries: Thurston’s three most nontrivial geometries.

Twisted Product Geometries

In dimension three, according to the famous Thurston geometrization conjecture, there are eight model geometries, which are simple connected, complete, homogeneous Riemannian manifolds.

We explore Nil, SL2(R), and Sol spaces — the three most nontrivial Thurston geometries. These geometries are examples of Lie groups, which are manifolds that admit a group structure.

Nil Space

The Nil geometry is a R-bundle over R^2. This geometry is fully carried by a Lie Group called the Heisenberg group.

SL2(R) Space

The SL2(R) geometry is similar to Nil, but it is a R-Bundle over H^2. The geometry is fully carried by the Lie group SL2, which is the universal cover of SL2(R) and PSL2(R). It is convenient to identify PSL2(R) with the unit tangent bundle of H^2.

Sol Space

The Sol geometry is the least symmetric of the eight Thurston Geometries. It has a bundle structure, but with a one-dimensional basis: it is a R^2-bundle over R. The geometry is fully governed by the Lie group Sol.