A Geometric Method for Determining Intersection Relations Between a Movable Convex Object and a Set of Planar Polygons

In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon A and a set Φ of possibly overlapping polygons fixed in the plane. More specifically, a subset Φ ⊂ Ξ is called an intersection relation if there exists a placement of A that intersects, and only intersects, Φ. The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all of the intersection relations between A and Ξ. Past related research only focuses on the complexity of the free space of the configuration space between A and Ξ and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space, but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations.