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Finding the distance between two circles in three-dimensional space

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1 Author(s)
C. A. Neff ; IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA

In this paper we investigate, from an algebraic point of view, the problem of finding the distance between two circles located in ℝ3 . We show, by combining a theorem about solvable permutation groups and some explicit calculations with a computer algebra system, that, in general, the distance between two circles is an algebraic function of the parameters defining them, but that this function is not solvable in terms of radicals. Although this result implies that one cannot find a “closed-form” solution for the distance between an arbitrary pair of circles in ℝ3, we discuss how such an algebraic quantity can still be manipulated symbolically by combining standard polynomial operations with an algorithm for isolating the real roots of a polynomial in a convenient data structure for real algebraic numbers. This data structure and its operations have been implemented.

Note: The Institute of Electrical and Electronics Engineers, Incorporated is distributing this Article with permission of the International Business Machines Corporation (IBM) who is the exclusive owner. The recipient of this Article may not assign, sublicense, lease, rent or otherwise transfer, reproduce, prepare derivative works, publicly display or perform, or distribute the Article.  

Published in:

IBM Journal of Research and Development  (Volume:34 ,  Issue: 5 )