| Summary: | 碩士 === 國立高雄第一科技大學 === 電腦與通訊工程所 === 96 === Determining the minimum distance between two objects is an important problem in robotics, image processing, CAD systems, pattern recognition, computational geometry and other areas of information processing which deal with geometrical data. For instance, one of the most classical applications related to this problem is the “path-planning” problem. All work that is developed for automated path planning requires at its lowest level the ability to detect whether or not collision has occurred. The ability to compute distance efficiently will result in a substantial reduction in the overall time required for most path-planning algorithms. In general, a path planning algorithm needs to ascertain for any position in the workspace not only if a collision has occurred, but also how close it is to occurring if it has not.
Until recently, the majority of the results obtained have been restricted to the minimum distance problem between a small class of geometric objects: points, lines, line segments, polygons and polyhedra. Despite the extensive algorithms and algorithmic techniques for objects defined with straight edges and flat faces, few of their results apply directly to objects of the real world, in general with curved shapes. Instead, the way to tackle arbitrary real objects has been to approximate them first as polygons or polyhedra of a sufficient number of vertices for the particular application. This process is generally quite unsatisfactory.
In this dissertation, we will restrict our studies to the minimum distance problem between two simple planar circular hulls, i.e., those circular hulls whose boundarys do not intersect themselves and whose vertices are represented by ordered pairs, in either Cartesian coordinates or polar coordinates. Circular hulls are similar to ordinary convex hull, except their edges are arcs of circles. They play an important role as a component of a large number of more complex problems. The minimum distance problem between two circular hulls has many important applications such as transportation and locating stations and facilities, one of which is the establishing of radio stations, medical centers and fire stations.
The main contribution of our work is the algorithm which solves the minimum distance problem between two circular hulls in O(logn) time, equivalent to the case of convex polygons and we strongly believed it is optimal for solving this problem. In addition, we also introduce the notion of circular hulls and establish some geometric properties, especially the relationships between the circular hull and the outer nearest-neighbor diagram of a point set, which may be useful in solving a variety of geometric problems.
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