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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