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Issue No.06 - November/December (2002 vol.22)
pp: 90-99
Jeff Weeks , MacArthur Fellow
ABSTRACT
<p>Curved spaces offer new opportunities to artists and game developers. The surface of a hypersphere provides a pleasantly finite 3D world in which the user can fly freely without ever encountering a boundary of any sort, while its negatively curved cousin hyperbolic space provides a startlingly spacious environment. Rendering perspectively correct images in curved spaces is easy, because the basic algorithms are identical to the standard flat space algorithm. When given the correct transformation matrices, standard 3D graphics cards render scenes in curved spaces as quickly and accurately as they do in flat space. Sample code in OpenGL and Direct3D provides a starting point for readers wishing to develop their own curved space applications.</p>
INDEX TERMS
curved space, hypersphere, 3-sphere, hyperbolic space, graphics, rendering, homogeneous coordinates
CITATION
Jeff Weeks, "Real-Time Rendering in Curved Spaces", IEEE Computer Graphics and Applications, vol.22, no. 6, pp. 90-99, November/December 2002, doi:10.1109/MCG.2002.1046633
REFERENCES
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5. M. Phillips and C. Gunn, "Visualizing Hyperbolic Space: Unusual Uses of 4x4 Matrices," Proc. 1992 Symp. Interactive 3D Graphics, ACM Press, New York, 1992, pp. 209-214.
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