Computational geometry: an introduction
Computational geometry: an introduction
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
A computational solution of the inverse problem in radiation-therapy treatment planning
Applied Mathematics and Computation
Topologically sweeping an arrangement
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Arrangements of curves in the plane—topology, combinatorics, and algorithms
Theoretical Computer Science
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Determining an optimal penetration among weighted regions in two and three dimensions
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Computing the arrangement of curve segments: divide-and-conquer algorithms via sampling
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Parallel Optimal Weighted Links
ICCS '01 Proceedings of the International Conference on Computational Sciences-Part I
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The problem of computing an optimal beam among weighted regions (called the optimal beam problem) arises in several applied areas such as radiation therapy, stereotactic brain surgery, medical surgery, geological exploration, manufacturing, and environmental engineering. In this paper, we present computational geometry techniques that enable us to develop efficient algorithms for solving various optimal beam problems among weighted regions in two and three dimensions. In particular, we consider two types of problems: the covering problems (seeking an optimal beam to contain a specified target region), and the piercing problems (seeking an optimal beam of a fixed shape to pierce the target region). We investigate several versions of these problems, with a variety of beam shapes and target region shapes in 2-D and 3-D. Our algorithms are based on interesting combinations of computational geometry techniques and optimization methods, and transform the optimal beam problems to solving a collection of instances of certain special non-linear optimization problems. Our approach makes use of interesting geometric observations, such as utilizing some new features of Minkowski sums.