An extension of Kim's algorithm for Apollonius tenth problem

The Kim's method is a computationally efficient and easy-to-code algorithm for the Apollonius tenth problem. However, the method does not analyze the positional relationship of three arbitrary circles and the desired Apollonius circles may be nonexistent. In this paper, we first classify common tangent lines of two circles according to the positional relationship of the two circles and judge the case the Apollonius circle does not exist. We further indicate which type of common tangent lines need to be computed and propose a geometric approach to calculate them. After that the common tangent lines are transformed into circles by inverse Möbius transformation and then the desired Apollonius circles are generated by using radius post-adjustment rule to dispose the circles. Experimental results show that our method can efficiently deal with arbitrary case of the Apollonius tenth problem.