Abstract
We prove that any conformal mapping between three-dimensional Cartesian spaces is limited to translation, rotation, uniform scaling, reciprocal, or simple combinations of these four operations. This result extends our earlier work on the CRV (Conformal Rotation Vector), which defines a conformal mapping from the non-Euclidean space of rigid body orientations into a Euclidean 3-space. Conformal means that changes in orientation resulting from two consecutive body rotations, about mutually-perpendicular axes through equal infinitesimal angles, are represented by mutually- perpendicular equal-sized vector increments in CRV space. This property makes the CRV particularly useful for motion planning in robots. Because the conformal mapping between CRV space and another Cartesian space is limited, the CRV is essentially unique.
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© 1994 Springer Science+Business Media Dordrecht
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Milenkovic, V., Milenkovic, P.H. (1994). Limited Existence of Three-Dimensional Conformal Mapping in Robots. In: Lenarčič, J., Ravani, B. (eds) Advances in Robot Kinematics and Computational Geometry. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8348-0_6
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DOI: https://doi.org/10.1007/978-94-015-8348-0_6
Publisher Name: Springer, Dordrecht
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