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

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

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

We introduce a method of calculating and rendering shapes in a non-Euclidean 2D space in real-time using hyperbolic and spherical trigonometry. We record the objects’ parameters in a polar coordinate system and use azimuthal equidistant projection to render the space onto the screen. We discuss the complexity of this method, renderings produced, limitations and possible applications of the created software as well as potential future developments

Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single "basic trigonometric group equation", and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.Comment: 46 pages, LaTe

Non-Euclidean Video Games: Exploring Player Perceptions and Experiences inside Impossible Spaces

Non-Euclidean geometry has the potential to be used for novel interactions in video games and create virtual spaces that are not physically possible in the real world. To explore how players perceive and experience them in video games, we have adapted two well-known 2D games, Snake and Asteroids to create two versions in addition to the conventional virtual space – with hyperbolic and spherical environments – and conducted a within-subject design user study on all three versions of these games. The results show that experienced Mastery and Control are lower when playing the two non-Euclidean versions while perceived Immersion and Challenge do not differ significantly between these three conditions. We also report on the qualitative findings from our participants, which provide further insights into the perception and experiences of these environments

Notes on Bolyai's 'Appendix'

This paper aims to provide an explanatory edition of Bolyai's 'Appendix Demonstrating the Absolute Science of Space', first published in 1832. In this treatise Bolyai began by extending neutral (or 'absolute') geometry by deriving a number of theorems which are independent of Euclid's parallel postulate. Then, while retaining Euclid first four postulates, he explored the consequences of replacing the parallel postulate with a different assumption, that through a given point not on a given straight line, more than than one straight line can be drawn which does not intersect the given line. On this basis Bolyai developed a non-Euclidean geometry nowadays called hyperbolic geometry. The English translation of Bolyai's text is reprinted in the paper, with explanatory notes provided at the end of each section. Some of the theorems are discussed in detail, with a particular focus on those in which Bolyai derived the trigonometric identities of the hyperbolic plane and on his astonishing quadrature of the circle. Bolyai's terse style of exposition can be quite challenging, so I have tried to fill in some of the mathematical details which he chose to omit, either through lack of space or simply because he thought them to obvious too include.Comment: 96 pages, 49 figure

Puzzling the 120-cell

We introduce Quintessence: a family of burr puzzles based on the geometry and combinatorics of the 120-cell. We discuss the regular polytopes, their symmetries, the dodecahedron as an important special case, the three-sphere, and the quaternions. We then construct the 120-cell, giving an illustrated survey of its geometry and combinatorics. This done, we describe the pieces out of which Quintessence is made. The design of our puzzle pieces uses a drawing technique of Leonardo da Vinci; the paper ends with a catalogue of new puzzles.Comment: 25 pages, many figures. Exposition and figures improved throughout. This is the long version of the shorter published versio

Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry

Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession

Lobachevski Illuminated: Content, Methods, and Context of the Theory of Parallels

In the 1820\u27s, Nikolai Ivanovich Lobachevski discovered and began to explore the world\u27s first non-Euclidean geometry. This crucial development in the history of mathematics was not recognized as such in his own lifetime. When his work finally found a sympathetic audience in the late 19th century, it was reinterpreted in the light of various intermediate developments (particularly Riemann\u27s conception of geometry), which were foreign to Lobachevski\u27s own way of thinking about the subject. Because our modern understanding of his work derives from these reinterpretations, many of Lobachevski\u27s most striking ideas have been forgotten. To recover them, I have produced an illuminated version of Lobachevski\u27s most accessible work, Geometrische Untersuchungen zur Theorie der Parallellinien (Geometric Investigations on the Theory of Parallels), a book that he published in 1840. I have produced a new English version of this work, together with extensive mathematical, historical, and philosophical commentary. The commentary expands and explains Lobachevski\u27s often cryptic statements and proofs, while linking the individual propositions of his treatise to the related work of his predecessors (including Gerolamo Saccheri, J.H. Lambert, and A.M. Legendre), his contemporaries (including J·nos Bolyai and Karl Friedrich Gauss), and his followers (including Eugenio Beltrami, Henri PoincarÈ, and David Hilbert). This dissertation supplies the contemporary reader with all of the tools necessary to unlock Lobachevski\u27s rich, beautiful, but generally inaccessible world