New upper bounds in Klee's measure problem
SIAM Journal on Computing
ACM Transactions on Graphics (TOG)
Scaling and related techniques for geometry problems
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Finding an optimal path without growing the tree
Journal of Algorithms
Computing the volume of the union of cubes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Five-axis tool path generation for a flat-end tool based on iso-conic partitioning
Computer-Aided Design
Free-Form Surface Partition in 3-D
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Image Segmentation with Monotonicity and Smoothness Constraints
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
Data structures for range minimum queries in multidimensional arrays
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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We present several algorithms for computing a feasible toolpath of certain characteristics for sculpting a given surface using a 5-axis numerically controlled (NC) machine. A toolpath specifies the orientations of a cutting tool at each point of a path taken by the tool. When a single toolpath does not exist, we find the minimum number of toolpaths needed by the cutting tool. Previous algorithms are all heuristics with no quality guarantee of a solution and with no analysis of the running time. We obtain optimal solutions and provide time analysis for all our algorithms. We model the problem using a directed, layered graph G (representing the sculpting constraints) such that a feasible toolpath corresponds to a certain path in G. We give efficient methods for several path problems in such graphs (e.g., finding a path in an unweighted or vertex-weighted version of G, computing the minimum number of paths whose union spans all the layers of an edge-weighted G, etc).