Deriving the Minimum Traversal Function of Three Circles

The technique of layered manufacturing in rapid prototyping is to fabricate product prototype by scanning the cross-sectional contours of the product using a laser beam layer by layer. The outlines of geometrical objects on each layer are different, and each layer may contain several geometrical objects. In order to simplify the problem, each geometrical object on the same plane is approximating by its own minimum circumscribed circle. Therefore, the minimum traversal path between circles can be the based model of the minimum traversal path between scanning geometrical objects. Furthermore, the minimum traversal path of three circles is the degenerated case of the minimum traversal path between circles. And the problem of the minimum traversal path of three circles can be transferred to the problem of the minimum traversal path of one circle and two points in this research. By using the concepts of reflection of light in physics and geometrical mathematics, the equation of the minimum traversal path between three circles will be derived in this paper. This equation of minimum traversal path can be applied in many application areas, including robotic motion planning and path planning for submarine, ship, and airplane.