Apollonius tenth problem via radius adjustment and Möbius transformations

Authors:
Donguk Kim;Deok-Soo Kim;Kokichi Sugihara
Affiliations:
Voronoi Diagram Research Center, Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791 Korea;Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791 Korea;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7 3 1, Hongo, Bunkyo-ku, Tokyo 113-0033 Japan
Venue:
Computer-Aided Design
Year:
2006

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Abstract

The Apollonius Tenth Problem, as defined by Apollonius of Perga circa 200 B.C., has been useful for various applications in addition to its theoretical interest. Even though particular cases have been handled previously, a general framework for the problem has never been reported. Presented in this paper is a theory to handle the Apollonius Tenth Problem by characterizing the spatial relationship among given circles and the desired Apollonius circles. Hence, the given three circles in this paper do not make any assumption regarding on the sizes of circles and the intersection/inclusion relationship among them. The observations made provide an easy-to-code algorithm to compute any desired Apollonius circle which is computationally efficient and robust.