Immersive Visualization of Classical Non-Euclidean Spaces
We have developed an experimental platform for immersive visualization of the Classical Non-Euclidean Spaces using real-time ray tracing.
The project includes the design and implementation of an extensive framework for creating interactive experiences in landscapes of three-dimensional Manifolds / Orbifolds having their geometric structure modeled by one of the classical tridimensional geometries, i.e., Flat (E3), Hyperbolic (H3), and Spherical (S3).
Geometric manifolds are abstract spaces spaces locally similar to the Euclidean space. We present the three classical examples of such spaces: Euclidean, hyperbolic, and spherical spaces.
The 3-dimensional torus T3 is generated by the action of the group of translations in the Euclidean space which coves T3, explaining thus the copies pattern.
Seifert-Weber Dodecahedral Space
Considering the dodecahedron embedded in the hyperbolic space, a special clockwise identification of its opposite faces gives rise to Seifert-Weber dodecahedral space. Its geometric structure is modelled by H3.
Poincaré Dodecahedron Space
Discovered by Poincare, is obtained by gluing the opposite faces of a dodecahedron embedded in the spherical space. Its geometric structure is modelled by S3.
Such spaces are modeled locally by quotients of a model geometry by discrete groups. We present two simple orbifold examples : the mirrored cube, and the mirrored dodecahedron.
The mirrored cube is an example of a non-manifold with the geometric structure modeled by E3 through a special group of reflections.
For an example of a non-manifold with geometric structure modeled by the hyperbolic space, consider the dodecahedron embedded in H3. Let T be the group of reflections generated by the dodecahedral faces. The quotient H3/T is the mirrored dodecahedral space.