The power of geometric duality
BIT - Ellis Horwood series in artificial intelligence
Computational geometry: an introduction
Computational geometry: an introduction
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Polyhedral line transversals in space
Discrete & Computational Geometry - ACM Symposium on Computational Geometry, Waterloo
Computer numerical control of machine tools
Computer numerical control of machine tools
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
Oriented projective geometry
Algorithmic aspects of alternating sum of volumes. Part 2: Nonvergence and its remedy
Computer-Aided Design
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
Primitives for the manipulation of general subdivisions and the computation of Voronoi
ACM Transactions on Graphics (TOG)
Computational Geometry for Design and Manufacture
Computational Geometry for Design and Manufacture
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Geometric transforms for fast geometric algorithms
Geometric transforms for fast geometric algorithms
Arbitrarily precise computation of Gauss maps and visibility sets for freeform surfaces
SMA '95 Proceedings of the third ACM symposium on Solid modeling and applications
Proceedings of the sixth ACM symposium on Solid modeling and applications
Accessibility Analysis Using Computer Graphics Hardware
IEEE Transactions on Visualization and Computer Graphics
Computing an exact spherical visibility map for meshed polyhedra
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Approximating centroids for the maximum intersection of spherical polygons
Computer-Aided Design
Characterization of polyhedron monotonicity
Computer-Aided Design
Computing feasible toolpaths for 5-axis machines
Theoretical Computer Science
Global obstacle avoidance and minimum workpiece setups in five-axis machining
Computer-Aided Design
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We consider the computation of an optimal workpiece orientation allowing the maximal number of surfaces to be machined in a single setup on a three-, four-, or five-axis numerically controlled machine. Assuming the use of a ball-end cutter, we establish the conditions under which a surface is machinable by the cutter aligned in a certain direction, without the cutter's being obstructed by portions of the same surface. The set of such directions is represented on the sphere as a convex region, called the visibility map of the surface. By using the Gaussian maps and the visibility maps of the surfaces on a component, we can formulate the optimal workpiece orientation problems as geometric problems on the sphere. These and related geometric problems include finding a densest hemisphere that contains the largest subset of a given set of spherical polygons, determining a great circle that separates a given set of spherical polygons, computing a great circle that bisects a given set of spherical polygons, and finding a great circle that intersects the largest or the smallest subset of a set of spherical polygons. We show how all possible ways of intersecting a set of n spherical polygons with v total number of vertices by a great circle can be computed in O(vn log n) time and represented as a spherical partition. By making use of this representation, we present efficient algorithms for solving the five geometric problems on the sphere.