Abstract
Conformal transformations are described by rotors in the conformal model of geometric algebra (CGA). In applications there is a need for interpolation of such transformations, especially for the subclass of 3D rigid body motions. This chapter gives explicit formulas for the square root and the logarithm of rotors in 3D CGA. It also classifies the types of conformal transformations and their orbits. To derive the results, we employ a novel polar decomposition for the even subalgebra of 3D CGA and an associated norm-like expression.
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It is the inversion of a uniform scaling with respect to an inversion sphere that has one of the points of B at its center and the other on its shell; dually it is \((B \pm\sqrt{B^{2}}) (n_{\infty}\rfloor B) + 2 B^{2} n_{\infty}\), with n ∞ the point at infinity.
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Dorst, L., Valkenburg, R. (2011). 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. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_5
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DOI: https://doi.org/10.1007/978-0-85729-811-9_5
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