Design and visualization of Riemannian metrics

TL;DR

This paper presents methods for constructing and visualizing Riemannian metrics in three-dimensional space to generate special effects like warping and mirages, using graph functions, diffeomorphisms, and real-time GPU-based ray tracing.

Contribution

It introduces new techniques for Riemannian metric construction in r^3, including geodesic derivation and real-time visualization with GPU acceleration.

Findings

Effective visualization of Riemannian effects in real-time

Novel methods for metric construction using graphs and diffeomorphisms

Demonstration of special effects like warping and mirages

Abstract

Local and global illumination were recently defined in Riemannian manifolds to visualize classical Non-Euclidean spaces. This work focuses on Riemannian metric construction in $\mathbb{R}^3$ to explore special effects like warping, mirages, and deformations. We investigate the possibility of using graphs of functions and diffeomorphism to produce such effects. For these, their Riemannian metrics and geodesics derivations are provided, and ways of accumulating such metrics. We visualize, in "real-time", the resulting Riemannian manifolds using a ray tracing implemented on top of Nvidia RTX GPUs.