Abstract
Layered manufacturing in rapid prototyping is to fabricate prototypes by using a laser beam to trace the cross-sectional contours of a product layer by layer. Such cross-sections of geometrical objects differ by layers and generally have more than one continuous contour in each layer. In an attempt to facilitate an efficient approach for path planning, the problem is simplified by approximating each of the continuous contours with its minimum circumscribed circle. The tool path planning for traversing all the contours in the same cross-section can then be simplified as the path planning of circles. Finding the shortest path for three circles is fundamental to solving the more general problem. In this paper, the problem of finding the minimum traversal path of three circles is transferred to the problem of finding the minimum traversal path of one circle and two points. By using the concepts of light reflection and accompanied by geometric mathematics, the equation of the minimum traversal path of three circles is derived. By analyzing the initially obtained eight roots, the two-root solution function is derived. This solution function can be used for applications including robotic motion planning and path planning for submarines, ships, and airplanes.