Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds

TL;DR

This paper presents an interactive system for designing and visualizing arbitrary non-Euclidean kaleidoscopic orbifolds using mirror rooms, M"obius transformations, and adapted rendering algorithms for hyperbolic and spherical geometries.

Contribution

It introduces a novel algorithm to construct polygons matching any 2D orbifold and enables interactive visualization of non-Euclidean spaces with reflection-based scenes.

Findings

Successfully visualizes non-Euclidean orbifolds in real-time

Allows interactive editing of kaleidoscopic scenes

Provides insights into orbifold notation and universal covers

Abstract

Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds. Given any arbitrary two-dimensional kaleidoscopic orbifold, we provide an algorithm to construct a Euclidean, spherical, or hyperbolic polygon to match the orbifold. This polygon is then used to create a room for which the polygon serves as the floor and the ceiling. With our system that implements M\"obius transformations, the user can interactively edit the scene and see the reflections of the edited objects. To correctly visualize non-Euclidean orbifolds, we adapt the rendering algorithms to account for the geodesics in these spaces, which light rays follow. Our interactive orbifold design system allows the user to create arbitrary two-dimensional kaleidoscopic orbifolds. In addition, our mirror-based orbifold visualization approach has the potential of helping our users gain insight on the orbifold, including its orbifold notation as well as its universal cover, which can also be the spherical space and the hyperbolic space.