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On the algebraic and geometric foundations of computer graphics

Author:

Ron GoldmanAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 21, Issue 1

Pages 52 - 86

https://doi.org/10.1145/504789.504792

Published: 01 January 2002 Publication History

Abstract

Today's computer graphics is ostensibly based upon insights from projective geometry and computations on homogeneous coordinates. Paradoxically, however, projective spaces and homogeneous coordinates are incompatible with much of the algebra and a good deal of the geometry currently in actual use in computer graphics. To bridge this gulf between theory and practice, Grassmann spaces are proposed here as an alternative to projective spaces. We establish that unlike projective spaces, Grassmann spaces do support all the algebra and geometry needed for contemporary computer graphics. We then go on to explain how to exploit this algebra and geometry for a variety of applications, both old and new, including the graphics pipeline, shading algorithms, texture maps, and overcrown surfaces.

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Index Terms

On the algebraic and geometric foundations of computer graphics
1. Computing methodologies
  1. Artificial intelligence
    1. Computer vision
      1. Computer vision representations
        Hierarchical representations
  2. Computer graphics
    1. Shape modeling

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 21, Issue 1

January 2002

86 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/504789

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Copyright © 2002 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 2002

Published in TOG Volume 21, Issue 1

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Cited By

Xiao TDeng JWen CGu Q(2024)Parallel algorithm for multi-viewpoint viewshed analysis on the GPU grounded in target cluster segmentationInternational Journal of Digital Earth10.1080/17538947.2024.230870717:1Online publication date: 25-Jan-2024
https://doi.org/10.1080/17538947.2024.2308707
Lu BZhou JZhang YLiu YWang Q(2024)An alternative rotating object detection method for rock particle size distribution analysisPowder Technology10.1016/j.powtec.2024.120059(120059)Online publication date: Jul-2024
https://doi.org/10.1016/j.powtec.2024.120059
Wu ZYang R(2024)The Gauss-cos model for the autocorrelation function of fertility rateApplied Mathematics and Computation10.1016/j.amc.2024.128907480(128907)Online publication date: Nov-2024
https://doi.org/10.1016/j.amc.2024.128907
Garnier LBécar JDruoton LFuchs LMorin G(2024)Theory of Minkowski-Lorentz SpacesEncyclopedia of Computer Graphics and Games10.1007/978-3-031-23161-2_111(1862-1878)Online publication date: 5-Jan-2024
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Vaz JMann S(2020)On the Clifford Algebraic Description of Transformations in a 3D Euclidean SpaceAdvances in Applied Clifford Algebras10.1007/s00006-020-01080-w30:4Online publication date: 2-Aug-2020
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Ghali S(2018)Sense and sidedness in the graphics pipeline via a passage through a separable spaceThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-008-0302-425:4(367-375)Online publication date: 27-Dec-2018
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Vaz JMann S(2018)Paravectors and the Geometry of 3D Euclidean SpaceAdvances in Applied Clifford Algebras10.1007/s00006-018-0916-128:5Online publication date: 3-Nov-2018
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