[Paper Review] Non-Euclidean Virtual Reality IV: Sol

Rémi Coulon, Elisabetta A. Matsumoto+2 more|arXiv (Cornell University)|Feb 2, 2020

Mathematics and Applications5 references4 citations

TL;DR

This paper presents a real-time, immersive virtual reality simulation of Sol geometry, a non-Euclidean 3-manifold with a unique Riemannian metric. Using geodesic ray-marching and parallel transport via Lie group structure, the system enables users to explore intrinsic visual properties—such as asymmetric stretching of spheres and non-intuitive plane foliations—offering a geometrically accurate, interactive experience of Sol's hyperbolic and anisotropic features.

ABSTRACT

This article presents virtual reality software designed to explore the Sol geometry. The simulation is available on 3-dimensional.space/sol.html

Motivation & Objective

• To develop a real-time, immersive virtual reality environment for exploring Sol geometry, a non-Euclidean 3-manifold with anisotropic metric properties.
• To enable users to experience the intrinsic geometry of Sol through accurate ray-marching along geodesics, simulating how light propagates in the space.
• To implement a computationally efficient and geometrically correct rendering engine that supports user navigation and stereoscopic vision in Sol and its compact quotients.
• To address the challenge of simulating non-isotropic, non-symmetric geometries by leveraging the Lie group structure of Sol and numerical integration of geodesic flows.
• To provide a foundation for future research in visualizing homogeneous Riemannian spaces by extending prior work on hyperbolic and product geometries.

Proposed method

• The simulation uses a Riemannian metric $ ds^2 = e^{-2z}dx^2 + e^{2z}dy^2 + dz^2 $ on $ \mathbb{R}^3 $, defining the Sol geometry as a solvable Lie group with group law $ (x_1,y_1,z_1) * (x_2,y_2,z_2) = (e^{z_1}x_2 + x_1, e^{-z_1}y_2 + y_1, z_1 + z_2) $.
• Geodesics are computed by solving a system of second-order differential equations derived from the Levi-Civita connection: $ \ddot{x} = 2\dot{x}\dot{z} $, $ \ddot{y} = -2\dot{y}\dot{z} $, $ \ddot{z} = -e^{-2z}(\dot{x}^2 - \dot{y}^2) $.
• The system employs Runge-Kutta numerical integration to simulate geodesic flow and ray-marching, with teleportation via discrete group actions $ \Gamma $ to model compact quotient manifolds.
• Parallel transport along geodesics is implemented using the matrix $ Q(t) $ satisfying $ \dot{Q} + BQ = 0 $, where $ B $ encodes the tangent vector's components in the Lie algebra.
• User movement is modeled as geodesic motion in the direction of the viewer’s gaze, with orientation updated via parallel transport to preserve geometric fidelity.
• The rendering pipeline supports stereoscopic vision by computing two viewpoints simultaneously, enabling immersive VR experiences on headsets.

Experimental results

Research questions

• RQ1 How can non-Euclidean geometries like Sol be accurately and interactively visualized in real time using virtual reality?
• RQ2 What are the intrinsic visual properties of Sol geometry—such as the shape of geodesic spheres and the behavior of light rays—when viewed through a geodesic ray-marching algorithm?
• RQ3 How can parallel transport and isometric group actions be numerically implemented to maintain geometric consistency during user navigation in Sol space?
• RQ4 What are the visual and perceptual consequences of Sol’s anisotropic metric, such as the 'tube-like' distortion of horizontal planes when viewed from above?
• RQ5 How can compact quotient manifolds of Sol be simulated using fundamental domains and group actions to enable closed, finite virtual environments?

Key findings

• Geodesic spheres in Sol exhibit $ D_8 $-symmetry and are stretched along the $ y $-axis for $ z > 0 $ and along the $ x $-axis for $ z < 0 $, due to the metric’s exponential scaling in $ x $ and $ y $.
• Horizontal planes $ \{z = c\} $ are not totally geodesic; only specific geodesics like $ \gamma_{\pm}(t) = (t, \pm t, 0) $ lie entirely within them.
• When viewed from above, the $ xy $-plane appears 'rolled into a tube' due to the non-Euclidean geometry, with some geodesics making a 'u-turn' and re-entering the plane.
• Some light rays pointing downward from above the $ xy $-plane hit the plane, reverse direction, and re-enter it, creating the illusion of objects behind the plane.
• The simulation correctly models the absence of 1-parameter isometry groups for displacement and rotation, requiring careful integration of group actions and parallel transport.
• The system supports real-time navigation and stereoscopic rendering, though performance and numerical accuracy of geodesic integration remain areas for future improvement.