Apollonius tenth problem via radius adjustment and Möbius transformations
Introduction
Circa 200 B.C., Apollonius of Perga, a famous Greek mathematician, formulated a set of problems for finding circles tangent to three geometric objects, each of which may be either a point, a line, or a circle. Among ten possible combinations of the objects, the computation of circles tangent to three given circles is the most interesting and involved problem. This problem is called Apollonius Tenth Problem.
Apollonius Tenth Problem is frequently encountered in various geometric problems as one of the core computations. In the construction of Euclidean Voronoi diagram for circles in a plane [4], [5], [10], [12], [15], for example, the position of a Voronoi vertex can be computed by solving Apollonius Tenth Problem defined by three circle generators. Fig. 1(a) shows a simple example of Voronoi diagram defined by three circles. In this figure, the Voronoi vertex (black dot), which is defined as the equi-distant point to the three circles, is the center of the Apollonius circle (dashed circle). Considering the potential applications of Voronoi diagrams for various problems [7], [8], [16], the systematic treatment of this problem is important.
In the Euclidean Voronoi diagram, there are various types of Apollonius circles depending on the relative location of the circle. For example, Fig. 1(b) shows a situation where the generators are located outside the Apollonius circle, while Fig. 1(c) shows another situation where one of the three generators contains the Apollonius circle. Note that both situations require different methods for computing the corresponding Apollonius circle.
Computing Apollonius circles in a general configuration is not easy since the algebraic solution for the Apollonius Tenth Problem may consist of too many terms [6]. Hence, directly solving the problem would require a significant amount of computation, and the accumulated numerical error may be serious so that the robustness of the computation cannot be guaranteed.
Ever since the problem was defined, there have been several studies to solve it [1], [11], [14]. Recently, Rokne reported on an approach for a particular case of the Apollonius Tenth Problem using Möbius transformation in a complex plane [13], and Gavrilova reported on an analytic solution for a similar setting which involves trigonometric functions [3]. Kim et al. reported on an approach to characterize different cases of the problem and provided a solution using Möbius transformation followed by a point location problem.
In this paper, we present a theory which provides an algorithm to easily compute all Apollonius circle(s). In our approach, there are no restrictions on the configuration of the generators, except when two generators coincide. Hence, generators are allowed to intersect each other and a generator may even contain other generators. The algorithm mainly uses Möbius transformation in a complex plane. It turns out that the proposed algorithm can compute all possible Apollonius circles. In addition to the ease of implementation, the proposed algorithm is numerically robust and computationally efficient.
This paper is organized as follows. In the following section, we exhaustively enumerate all possible cases of Apollonius circle with respect to three generators. Section 3 explains the Möbius transformation, which is one of the main mathematical tools used for the development of the proposed theory. In Section 4, the pre- and post-processing for the adjustment of generators are presented so that an algorithm to compute the solutions can be devised. Section 5 illustrates a few example showing how to apply the rules used in the algorithm for various examples. Finally, we give concluding remarks in Section 6.
Section snippets
Configurations of Apollonius circles
An Apollonius circle is defined as a circle tangent to three given circles called generators, Gi=(ci, ri), where i=0,1, and 2 with a center ci=(xi, yi) and a radius ri. Without loss of generality, we can assume that circles are ordered by their radii, i.e. ri≤rj where i<j. Apollonius Tenth Problem is, then, how to find all such Apollonius circles for any configuration of three circle generators. The generator G0, with the smallest radius, is called the smallest generator.
We now provide a
Möbius transformation
To facilitate the computation of the Apollonius circles for three generators, we transform the original problem to one of finding tangent circles to two circles while the tangent circle is passing through a point. This transformation is done via adjusting (either enlarging or shrinking) the radii of given generators, and we call this a radius adjustment transformation. As it will be discussed later, the amount of radius adjustment corresponds to the radius of the smallest generator. Note that
Radius adjustment transformations
In this section, we explain how to transform the Apollonius circle problem to the problem of finding a circle(s) tangent to two circles while passing through a point. We will refer to this a tangent circle problem hereafter.
As described in the previous section, we have three spaces: O-plane, Z-plane, and W-plane. To get an appropriate transformation to the Z-plane, the radii of generators should be adjusted, either by enlarging or shrinking, and this radius adjustment is called a pre-adjustment
Examples
In this section, we will explain the proposed algorithm by illustrating a few examples. Suppose that a given configuration is OXΔ and the generators are Gi=(ci, ri), i=0, 1, and 2, where r0≤r1≤r2. According to Rule 1, X- and Δ-generators, G1 and G2, should be enlarged since the status value of the smallest generator is O. As a result, a passing point z0≡c0 and two radius adjusted circles, Z1=(c1, r1+r0) and Z2=(c2, r2+r0) are obtained. Then, Z1 and Z2 are transformed to W1 and W2 in the W-plane
Conclusions
In this paper, we have presented a unified framework to solve the Apollonius Tenth Problem, as defined by Apollonius of Perga circa 200 B.C., in a general setting. Hence, we do not make any assumption, in this paper, regarding on the sizes of circles and the intersection/inclusion relationship among them. Even though particular cases have been handled previously, a general framework for the problem has never been reported.
The theory first characterizes the spatial relationship among given
Acknowledgement
The first two authors were supported by the Creative Research Initiatives from the Ministry of Science and Technology in Korea, and the third author was supported by the Japanese Society for the Promotion of Science.
Donguk Kim is a research assistant professor in Voronoi Diagram Research Center at Hanyang University, Korea. He received his B.S., M.S. and Ph.D. degrees from Hanyang University in 1999, 2001 and 2004, respectively. His research interests include computational geometry, geometric modeling and their applications in the molecular biology.
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Donguk Kim is a research assistant professor in Voronoi Diagram Research Center at Hanyang University, Korea. He received his B.S., M.S. and Ph.D. degrees from Hanyang University in 1999, 2001 and 2004, respectively. His research interests include computational geometry, geometric modeling and their applications in the molecular biology.
Deok-Soo Kim is a professor in Department of Industrial Engineering, Hanyang University, Korea. Before he joined the university in 1995, he worked at Applicon, USA, and Samsung Advanced Institute of Technology, Korea. He received a B.S. from Hanyang University, Korea, an M.S. from the New Jersey Institute of Technology, USA, and a Ph.D. from the University of Michigan, USA, in 1982, 1985 and 1990, respectively. His current research interests mainly lie in the theory and applications of Voronoi diagram while he has been interested in various geometric problems.
Kokichi Sugihara received the B.Eng., M.Eng. and Dr.Eng. degrees in Mathematical Engineering from the University of Tokyo in 1971, 1973 and 1980, respectively. Since 1986 he has been at the Department of Mathematical Engineering and Information Physics of the University of Tokyo, and he is now a professor. His research interests include computational geometry, computer graphics and computer vision. He is a member of the Information Processing Society of Japan, Operational Research Society of Japan, Japan SIAM, IEEE and ACM.