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

Jinta Zheng, Eugene Zhang, Yue Zhang

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2023-10-26T05:33:00ZGMT-0600Change your timezone on the schedule page
Exemplar figure, described by caption below
In this paper, we provide an algorithm to generate arbitrary two-dimensional non-Euclidean kaleidoscopic orbifolds. The example shown in this figure is a hyperbolic orbifold with six sides. The orders of symmetries at the six corners are respectively 2, 3, 4, 5, 6, and 7. Note that at a corner of order N, there are 2N sectors. Notice the reflections of the words "Non-Euclidean Orbifold" in the universal cover of the orbifold, the disk. The polygonal layout generated from our algorithm can be used as the ceiling and floor of a 3D room, thus enabling our mirror-based visualization for kaleidoscopic orbifolds.
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Kaleidoscopic Orbifolds, Orbifold Visualization, Math Visualization, Orbifold Construction, Spherical Geometry, Hyperbolic Geometry


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 Möbius 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.