Touring a Sequence of Spheres in R3

Abstract

How do you touring a sequence of different size of balls and returning to the starting point with a minimum Euclidean travelling path? We present an efficient algorithm to answer the question in this paper. One may say that the above mentioned minimum path, a shortest n‒gon of n‒spheres in three dimensions. Accompanied by three other competing methods: the genetic algorithm, the nearest neighbor algorithm and the random test, all have shown up our proposed algorithm dominates the results in every aspects of spheres distribution. Empirical results have also shown the effectiveness and the quick convergent property about the proposed algorithm. The proposed algorithm can be conducted for the coverage problem of sensor deployment. In addition, it is also called the shortest n‒gon of n‒spheres (particles with necessary volume). Since, practitioners in the fields of computational geometry, material science, solid modeling, 3D printing, computer graphics, or any other kinds of CAD/CAM applications may find merits from this paper.