A new polygon decomposition problem, the anchored area partition problem, which has applications to a multiple-robot terrain-covering problem is presented. This problem concerns dividing a given polygon V into n polygonal pieces, each of a specified area and each containing a certain point (site) on its boundary or in its interior. First the algorithm for the case when V is convex and contains no holes is presented. Then the generalized version that handles nonconvex and nonsimply connected polygons is presented. The algorithm uses sweep-line and divide-and-conquer techniques to construct the polygon partition. The input polygon V is assumed to have been divided into a set of p convex pieces (p = 1 when V is convex), which can be done in 0(vp log log up) time, where vp is the number of vertices of V and p = 0(vp), using algorithms presented elsewhere in the literature. Assuming this convex decomposition, the running time of the algorithm presented here is 0(pn 2 -run), where v is the sum of the number of vertices of the convex pieces.
An efficient, on-line terrain-covering algorithm is presented for a robot (AUV) moving in an unknown three-dimensional underwater environment. Such an algorithm is necessary for producing mosaicked images of the ocean floor. The basis of this three-dimensional motion planning algorithm is a new planar algorithm for nonsimply connected areas with boundaries of arbitrary shape. We show that this algorithm generalizes naturally to complex three-dimensional environments in which the terrain to be covered is projectively planar. This planar algorithm represents an improvement over previous algorithms because it results in a shorter path length for the robot and does not assume a polygonal environment. The path length of our algorithm is shown to be linear in the size of the area to be covered; the amount of memory required by the robot to implement the algorithm is linear in the size of the description of the boundary of the area. An example is provided that demonstrates the algorithm's performance in a nonsimply connected, nonplanar environment.
Abstract. We give an exact geometry kernel for conic arcs, algorithms for exact computation with low-degree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces (= the set of points where a linear or quadratic function is non-negative) by regularized boolean operations. The algorithm and its implementation are complete (they can handle all cases), exact (they give the mathematically correct result), and efficient (they can handle inputs with several hundred primitives).
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