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Rigid Body Dynamics and Conformal Geometric Algebra Read chapter Read first chapter

2011 | OriginalPaper | Chapter

4. Reconstructing Rotations and Rigid Body Motions from Exact Point Correspondences Through Reflections

Authors : Daniel Fontijne, Leo Dorst

Published in: Guide to Geometric Algebra in Practice

Publisher: Springer London

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Abstract

We describe a new algorithm to reconstruct a rigid body motion from point correspondences. The algorithm works by constructing a series of reflections which align the points with their correspondences one by one. This is naturally and efficiently implemented in the conformal model of geometric algebra, where the resulting transformation is represented by a versor. As a direct result of this algorithm, we also present a very compact and fast formula to compute a quaternion from two vector correspondences, a surprisingly elementary result which appears to be new.

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previous chapter Inverse Kinematics Solutions Using Conformal Geometric Algebra
next chapter Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra Using Polar Decomposition
Footnotes
1
When applied within the conformal model, (4.2) can find pure rotations and pure translations, but no general rigid body motions which require a grade-4 part in their versor.
 
2
An extension of this method to n-D (with n>6) may moreover be problematic since the Lagrangian constraints for the versor manifold are not yet known in general.
 
3
All benchmarks were performed on a 2.8-GHz Intel Core2Duo processor using 64-bit (double) floating point precision.
 
4
Editorial note: Chapter 7 gives a related method designed for redundant data.
 
Literature
1.
Aragon-Gonzalez, G., Aragon, J., Rodriguez-Andrade, M., Verde-Star, L.: Reflections, rotations, and Pythagorean numbers. Adv. Appl. Clifford Algebras 19, 1–14 (2009) MathSciNetMATHCrossRef
2.
Cibura, C., Dorst, L.: From exact correspondence data to conformal transformations in closed form using Vahlen matrices. In: Proceedings of GraVisMa 2009, Plzen
3.
4.
Dorst, L.: Determining a versor in n-D geometric algebra from the known transformation of n vectors. In: Proceedings of GraVisMa 2009, Plzen
5.
Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object Oriented Approach to Geometry, revised edn. Morgan Kaufmann, San Mateo (2009)
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7.
Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1997)
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Hanson, A.J.: Visualizing Quaternions. Morgan Kaufmann, San Mateo (2006)
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Lasenby, J., Fitzgerald, W.J., Doran, C.J.L., Lasenby, A.N.: New geometric methods for computer vision. Int. J. Comput. Vis. 191–213 (1998)
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Li, H., Hestenes, D., Rockwood, A.: Generalized homogeneous coordinates for computational geometry. In: Geometric Computing with Clifford Algebras. Springer Series in Information Science (2001)
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Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Berlin (2008)
12.
Richard, A., Fuchs, L., Charneau, S.: An algorithm to decompose n-dimensional rotations into planar rotations. In: Computational Modeling of Objects Represented in Images. Lecture Notes in Computer Science, vol. 6026, pp. 60–71. Spinger, Berlin (2010) CrossRef
Metadata
Title
Reconstructing Rotations and Rigid Body Motions from Exact Point Correspondences Through Reflections
Authors
Daniel Fontijne
Leo Dorst
Copyright Year
2011
Publisher
Springer London
Book
Guide to Geometric Algebra in Practice

Print ISBN: 978-0-85729-810-2

Electronic ISBN: 978-0-85729-811-9

Copyright Year: 2011

DOI
https://doi.org/10.1007/978-0-85729-811-9_4

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