Parameterization of 3D Conformal Transformations in Conformal Geometric Algebra

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.