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

by   Jinta Zheng, et al.
Oregon State University

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.


Navigating Higher Dimensional Spaces using Hyperbolic Geometry

Higher-dimensional spaces are ubiquitous in applications of mathematics....

Rendering Non-Euclidean Geometry in Real-Time Using Spherical and Hyperbolic Trigonometry

This paper introduces a method of calculating and rendering shapes in a ...

Illustrations of non-Euclidean geometry in virtual reality

Mathematical objects are generally abstract and not very approachable. I...

Browser-based Hyperbolic Visualization of Graphs

Hyperbolic geometry offers a natural focus + context for data visualizat...

Hyperbolic Image Embeddings

Computer vision tasks such as image classification, image retrieval and ...

Spherical Graph Drawing by Multi-dimensional Scaling

We describe an efficient and scalable spherical graph embedding method. ...

Latent Space Network Modelling with Continuous and Discrete Geometries

A rich class of network models associate each node with a low-dimensiona...

Please sign up or login with your details

Forgot password? Click here to reset