Computational geometry: an introduction
Computational geometry: an introduction
Cascading divide-and-conquer: a technique for designing parallel algorithms
SIAM Journal on Computing
Efficient binary space partitions for hidden-surface removal and solid modeling
Discrete & Computational Geometry - Selected papers from the fifth annual ACM symposium on computational geometry, Saarbrücken, Germany, June 5-11, 1989
Surpassing the information theoretic bound with fusion trees
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Computer graphics (2nd ed. in C): principles and practice
Computer graphics (2nd ed. in C): principles and practice
Static analysis yields efficient exact integer arithmetic for computational geometry
ACM Transactions on Graphics (TOG)
New results on binary space partitions in the plane
Computational Geometry: Theory and Applications
Approximate data structures with applications
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
A Characterization of Ten Hidden-Surface Algorithms
ACM Computing Surveys (CSUR)
A procedure for generation of three-dimensional half-toned computer graphics presentations
Communications of the ACM
Topological Lower Bounds on Algebraic Random Access Machines
SIAM Journal on Computing
Binary Space Partitions for Line Segments with a Limited Number of Directions
SIAM Journal on Computing
On visible surface generation by a priori tree structures
SIGGRAPH '80 Proceedings of the 7th annual conference on Computer graphics and interactive techniques
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Computer Graphics with OpenGL
Deterministic sorting in O(nlog logn) time and linear space
Journal of Algorithms
Interactive Computer Graphics: A Top-Down Approach using OpenGL (4th Edition)
Interactive Computer Graphics: A Top-Down Approach using OpenGL (4th Edition)
High-Performance Rendering on Clusters of Workstations
GMAI '06 Proceedings of the conference on Geometric Modeling and Imaging: New Trends
Half-tone perspective drawings by computer
AFIPS '67 (Fall) Proceedings of the November 14-16, 1967, fall joint computer conference
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The visibility of a planar set of n disjoint line segments, arising from the scanline approach to rendering three-dimensional scenes, is one of the classic problems in computer graphics. In order to solve the problem quickly, many authors proposed binary space partitioning (BSP) as a preprocessing, possibly breaking up the input line segments so that visibility is determined in time linear in the number of resulting segments. Tóth [Discrete & Comput. Geometry 30, 1 pp. 3-16, 2003] demonstrated that a BSP may result in Ω(n log n/ log log n) line segments. We demonstrate that the time and space complexities of the problem are Θ(n log n) and Θ(n) respectively, under the algebraic RAM model of computation. Introducing a more realistic model, a RAM with arbitrary-precision rational arithmetics, a deterministic algorithm is given that solves the problem directly, without the need of preprocessing, in O(n log log n) time and O(n) space, regardless of the precision of the input data.