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Real-Time Physics and Graphics Engine for non-Euclidean Geometry using Spherical and Hyperbolic Trigonometry

Osudin, D. (2022). Real-Time Physics and Graphics Engine for non-Euclidean Geometry using Spherical and Hyperbolic Trigonometry. (Unpublished Doctoral thesis, City, University of London)


This thesis presents an implementation of a 2D non-Euclidean physics and graphics engine using spherical and hyperbolic trigonometry. The engine is capable of working with a 2D space of constant negative or positive curvature. It uses polar coordinates to record the parameters of the objects as well as an azimuthal equidistant projection to render the space onto the screen. A polar coordinate system works well with trigonometric calculations, due to the distance from the reference point (analogous to origin in Cartesian coordinates) being one of the coordinates by definition. Azimuthal equidistant projection is not a typical projection, used for neither spherical nor hyperbolic space, however one of the main features of the engine relies on it: changing the curvature of the world in real-time.

Any 2D shape can be created and used in the engine, not a pre-determined list of standard shapes. Shapes can be moved around the curved space via user input controls.

This thesis describes approaches to improve performance of the engine by analysing and subsequently attempting to reduce the time-complexity of the algorithm as well as parallelizing the calculations by performing them on a GPU in order to avoid a major bottleneck. Empirical tests were performed and it was found that different approaches have an impact on overall engine performance, but the improvement is negligible compared to that gained by parallelisation.

A method for texturing shapes in non-Euclidean 2D space in real-time using spherical and hyperbolic trigonometry is introduced. Stress test results show that the engine can render high load scenes in real-time.

This thesis presents survey results showing participants’ generally positive feedback upon playing through two different classic games modified to work within the non-Euclidean engine.

Overall, the project has been successful in developing a novel method of rendering non-Euclidean geometry in real-time using Spherical and Hyperbolic trigonometry; implementing it within a framework which allows the creation of custom environment; and gauging the interest in non-Euclidean games.

Publication Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Q Science > QA Mathematics > QA76 Computer software
Departments: School of Science & Technology > Computer Science
School of Science & Technology > School of Science & Technology Doctoral Theses
Doctoral Theses
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