Ray tracing can also take care of teleporting at fundamental domain boundaries. Visualizing special and general relativity or black holes are good examples for these approaches. sphere tracing, is used instead, non-isotropic Thurston geometries or even geometries with varying curvature can also be rendered. If ray marching or its improved versions like, e.g. In ray tracing, only the ray definition and the intersection calculation should be altered according to the shortest path, i.e. image space rendering, or object space approaches. Visibility determination in 3D spaces can be based on ray tracing, i.e. Rendering non-Euclidean geometries is a topic in mathematics, cartography and art. 7, we examine the possibilities of game engine integration and show the shader programs of one particular solution. Section 6 discusses the adaptation of illumination and physics. Section 5 discusses how to convert objects obtained in the framework of Euclidean geometry into non-Euclidean spaces and introduces our solution based on the exponential map. 4 that presents formulas to build up the transformation matrices of typical operations in the animation and rendering pipeline. 3, we summarize the embedding space model of Euclidean, elliptic, hyperbolic and projective geometries. The structure of the paper is as follows: Section 2 surveys the previous work. However, it does not mean that the change of the curvature cannot be demonstrated since the same effect can be achieved by scaling Euclidean objects and locations before conversion. set the unit of length to the constant curvature of the space. Review of the possibilities of existing game engine adaptation. Modification of the laws of light propagation, illumination, and dynamics for curved spaces. ) The contributions are:Ī general framework and simple formulas with proofs to set up the transformation matrices according to the rules of elliptic and hyperbolic geometries.Ī method of converting game objects and worlds from Euclidean to non-Euclidean geometries. (Initial concepts related to the transformations in elliptic geometry were discussed in our previous paper. The objective of this paper is to convert graphics engines developed for Euclidean geometry to virtual worlds defined by elliptic or hyperbolic geometry. Unit圓D, implementing formulas assuming Euclidean geometry. These tasks are solved in game engines, e.g. Rendering the game objects determining the visibility and the radiance of their surfaces and projecting them onto the screen. Simulating game objects to determine their state including their translation and rotation in each frame.Īnimating game objects and the avatar’s camera by applying the computed transformations to vertices. Loading of the modeled objects into the game world. Games offer the experience of strange worlds thus, the adaptation of game engines to non-Euclidean geometries provides an interesting way of exploring and understanding these geometries.Ī game has to support the following main tasks: The difference in the parallel axiom has significant consequences thus, each of these geometries describes a different world. for a given line they postulate exactly one, none, and more than one non-intersecting line passing through a given point, respectively. Euclidean, elliptic and hyperbolic geometries share all but the parallel axiom, i.e.
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