Abstract
Conformal geometric algebra is a powerful mathematical language for describing and manipulating geometric configurations and their conformal transformations. By providing a 5D algebraic representation of 3D geometric configurations, conformal geometric algebra proves to be very helpful in pose estimation, motion design, and neuron-based machine learning (Bayro-Corrochano et al., J. Math. Imaging Vis. 24(1):55–81, 2006; Dorst et al., Geometric Algebra for Computer Science, Morgan Kaufmann, San Mateo, 2007; Hildenbrand, Comput. Graph. 29(5):795–803, 2005; Lasenby, Computer Algebra and Geometric Algebra with Applications, LNCS, vol. 3519, pp. 298–328, Springer, Berlin, 2005; Li et al., Geometric Computing with Clifford Algebras, pp. 27–60, Springer, Heidelberg, 2001; Mourrain and Stolfi, Invariant Methods in Discrete and Computational Geometry, pp. 107–139, Reidel, Dordrecht, 1995; Rosenhahn and Sommer, J. Math. Imaging Vis. 22:27–70, 2005; Sommer et al., Computer Algebra and Geometric Algebra with Applications, pp. 278–297, Springer, Berlin, 2005). In this chapter, we present some theoretical results on conformal geometric algebra which should prove to be useful in computer applications. The focus is on parameterizing 3D conformal transformations with either quaternionic Vahlen matrices or polynomial Cayley transform from the Lie algebra to the Lie group of conformal transformations in space.
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References
Ahlfors, L.V.: Möbius transformations in ℝn expressed through 2×2 matrices of Clifford numbers. Complex Var. 5, 215–224 (1986)
Angles, P.: Conformal Groups in Geometry and Spin Structures. Birkhäuser, Basel (2008)
Bayro-Corrochano, E., Reyes-Lozano, L., Zamora-Esquivel, J.: Conformal geometric algebra for robotic vision. J. Math. Imaging Vis. 24(1), 55–81 (2006)
Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann, San Mateo (2007)
Helmstetter, J., Micali, A.: Quadratic Mappings and Clifford Algebras. Birkhäuser, Basel (2000)
Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Kluwer, Dordrecht (1984)
Hildenbrand, D.: Geometric computing in computer graphics using conformal geometric algebra. Comput. Graph. 29(5), 795–803 (2005)
Lasenby, A.: Recent applications of conformal geometric algebra. In: Li, H., et al. (eds.) Computer Algebra and Geometric Algebra with Applications. LNCS, vol. 3519, pp. 298–328. Springer, Berlin (2005)
Li, H.: Invariant Algebras and Geometric Reasoning. World Scientific, Singapore (2008)
Li, H., Hestenes, D., Rockwood, A.: Generalized homogeneous coordinates for computational geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebras, pp. 27–60. Springer, Heidelberg (2001)
Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (1997)
Maks, J.: Clifford algebras and Möbius transformations. In: Micali, A., et al. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 57–63. Kluwer, Dordrecht (1992)
Mourrain, B., Stolfi, N.: Computational symbolic geometry. In: White, N.L. (ed.) Invariant Methods in Discrete and Computational Geometry, pp. 107–139. Reidel, Dordrecht (1995)
Riesz, M.: Clifford Numbers and Spinors, 1958. Kluwer, Dordrecht (1993). From lecture notes made in 1957–1958, edited by Bolinder, E. and Lounesto, P.
Rosenhahn, B., Sommer, G.: Pose estimation in conformal geometric algebra I, II. J. Math. Imaging Vis. 22, 27–70 (2005)
Ryan, J.: Conformal Clifford manifolds arising in Clifford analysis. Proc. R. Ir. Acad. A 85, 1–23 (1985)
Selig, J.M.: Geometrical Methods in Robotics. Springer, New York (1996)
Sommer, G., Rosenhahn, B., Perwass, C.: Twists—an operational representation of shape. In: Li, H., et al. (eds.) Computer Algebra and Geometric Algebra with Applications. LNCS, vol. 3519, pp. 278–297. Springer, Berlin (2005)
White, N.: Grassmann–Cayley algebra and robotics applications. In: Bayro-Corrochano, E. (ed.) Handbook of Geometric Computing, pp. 629–656. Springer, Heidelberg (2005)
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Li, H. (2010). Parameterization of 3D Conformal Transformations in Conformal Geometric Algebra. In: Bayro-Corrochano, E., Scheuermann, G. (eds) Geometric Algebra Computing. Springer, London. https://doi.org/10.1007/978-1-84996-108-0_4
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DOI: https://doi.org/10.1007/978-1-84996-108-0_4
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