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Generating Fractals Using Geometric Algebra

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Abstract

In this paper we investigate how, using the language of Geometric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended. In addition, via the use of the Conformal Model for Geometric Algebra, we develop an analog of complex arithmetic for the Poincaré disc and show that, in non-Euclidean geometries, there are two related but distinct variants of the Julia and Mandelbrot sets.

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References

  1. D. A. Brannan, M. F. Esplen and J. J. Gray, Geometry. Cambridge University Press, 1999, Ch. 6.

  2. Yumei Dang, Louis H. Kauffman and Daniel J. Sandin. Hypercomplex interations: distance estimation and higher dimensional fractals. World Scientific, 2002.

  3. D. Dunham, Transformation of Hyperbolic Escher Patterns. Visual Mathematics, 1 (1) 1999.

  4. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A unified language for mathematics and physics. Reidel, 1984.

  5. A. Lasenby, Recent applications of conformal geometric algebra. In International Workshop on Geometric Invariance and Applications in Engineering Xi’an, China, May 2004.

  6. A. Lasenby, J. Lasenby and R. Wareham, A Covariant Approach to Geometry using Geometric Algebra. Technical repot, Cambridge University Engineering Department, 2004. CUED/F-INFENG/TR-483.

  7. A. Rockwood, D. Hestenes, C. Doran, J. Lasenby, L. Dorst and S. Mann, Geometric Algebra. Course Notes. Course 31, Siggraph 2001, Los Angeles.

  8. Richard Wareham, Computer Graphics using Conformal Geometric Algebra. PhD thesis, University of Cambridge, 2006.

  9. Richard Wareham, Jonathan Cameron and Joan Lasenby, Applications of Conformal Geometric Algebra in Computer Vision and Graphics. In Hongbo Li, Peter J. Olver and Gerald Sommer, editors, Computer Algebra and Geometric Algebra with Applications: 6th International Workshop, IWMM 2004, Secaucus, NJ, USA, 2005. Springer-Verlag New York, Inc.

  10. Richard Wareham and Joan Lasenby. Mesh vertex pose and position interpolation using geometric algebra. In Proceedings of AMDO 2008, Mallorca, Spain, Secaucus, NJ, USA, 2008. Springer-Verlag New York, Inc.

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Authors and Affiliations

  1. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK

    R. J. Wareham & J. Lasenby

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  1. R. J. Wareham
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  2. J. Lasenby
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Correspondence to R. J. Wareham.

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Wareham, R.J., Lasenby, J. Generating Fractals Using Geometric Algebra. Adv. Appl. Clifford Algebras 21, 647–659 (2011). https://doi.org/10.1007/s00006-010-0265-1

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  • DOI: https://doi.org/10.1007/s00006-010-0265-1

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