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Ray-Tracing Objects and Novel Surface Representations in CGA

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Advances in Computer Graphics (CGI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11542))

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Abstract

Conformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds great promise in the field of computer graphics [1,2,3]. In this paper we implement a simple ray tracer in CGA with a Blinn-Phong lighting model and use it to examine ray intersections with surfaces generated from interpolating between objects [7]. An analytical method for finding the normal line to these interpolated surfaces is described. The expression is closely related to the concept of surface principal curvature from differential geometry and provides a novel way of describing the curvature of evolving surfaces.

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References

  1. Hestenes, D.: Old wine in new bottles: a new algebraic framework for computational geometry. In: Corrochano, E.B., Sobczyk, G. (eds.) Geometric Algebra with Applications in Science and Engineering. Birkhäuser, Boston (2001). https://doi.org/10.1007/978-1-4612-0159-5_1

    Chapter  Google Scholar 

  2. Lasenby, A., Lasenby, J., Wareham, R.: A Covariant Approach to Geometry Using Geometric Algebra., Cambridge University Engineering Department (2004)

    Google Scholar 

  3. Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry, 1st edn. Elsevier; Morgan Kaufmann, Amsterdam (2007)

    Google Scholar 

  4. Breuils, S., Nozick, V., Fuchs, L.: Garamon: geometric algebra library generator. In: AACA: Topical Collection AGACSE 2018, IMECC UNICAM, Campinas, Brazil (2018)

    Google Scholar 

  5. Hildenbrand, D.: Geometric computing in computer graphics using conformal geometric algebra. Comput. Graph. 29(5), 802–810 (2005)

    Article  Google Scholar 

  6. Wareham, R.J., Lasenby, J.: Generating fractals using geometric algebra. Adv. Appl. Clifford Algebras 21(3), 647–659 (2011)

    Article  MathSciNet  Google Scholar 

  7. Hadfield, H., Lasenby, J.: Direct linear interpolation of conformal geometric objects. In: AACA: Topical Collection AGACSE 2018, IMECC UNICAM, Campinas, Brazil (2018)

    Google Scholar 

  8. Dorst, L., Valkenburg, R.: Square root and logarithm of rotors in 3D conformal geometric algebra using polar decomposition. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice, pp. 81–104. Springer, London (2011). https://doi.org/10.1007/978-0-85729-811-9_5

    Chapter  MATH  Google Scholar 

  9. Blinn, J.: Models of light reflection for computer synthesized pictures. ACM SIGGRAPH Comput. Graph. 11(2), 192–198 (1977)

    Article  Google Scholar 

  10. Doran, C., Lasenby, A.: Geometric Algebra for Physicists, 1st edn. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Engineering, Cambridge University, Cambridge, UK

    Sushant Achawal, Joan Lasenby & Hugo Hadfield

  2. Department of Physics, Cambridge University, Cambridge, UK

    Anthony Lasenby

Authors
  1. Sushant Achawal
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  2. Joan Lasenby
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  3. Hugo Hadfield
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  4. Anthony Lasenby
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Corresponding author

Correspondence to Hugo Hadfield .

Editor information

Editors and Affiliations

  1. University of Calgary, Calgary, AB, Canada

    Marina Gavrilova

  2. Bournemouth University, Poole, UK

    Jian Chang

  3. University of Geneva, MIRALab, Carouge, Switzerland

    Nadia Magnenat Thalmann

  4. International Christian University, Tokyo, Japan

    Eckhard Hitzer

  5. Waseda University, Tokyo, Japan

    Hiroshi Ishikawa

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Achawal, S., Lasenby, J., Hadfield, H., Lasenby, A. (2019). Ray-Tracing Objects and Novel Surface Representations in CGA. In: Gavrilova, M., Chang, J., Thalmann, N., Hitzer, E., Ishikawa, H. (eds) Advances in Computer Graphics. CGI 2019. Lecture Notes in Computer Science(), vol 11542. Springer, Cham. https://doi.org/10.1007/978-3-030-22514-8_58

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  • DOI: https://doi.org/10.1007/978-3-030-22514-8_58

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22513-1

  • Online ISBN: 978-3-030-22514-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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