Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Applications of random sampling to on-line algorithms in computational geometry
Discrete & Computational Geometry
Voronoi diagrams and Delaunay triangulations
Handbook of discrete and computational geometry
Fast computation of generalized Voronoi diagrams using graphics hardware
Proceedings of the 26th annual conference on Computer graphics and interactive techniques
Computer Aided Geometric Design
Computing Voronoi skeletons of a 3-D polyhedron by space subdivision
Computational Geometry: Theory and Applications
Approximate medial axis as a voronoi subcomplex
Proceedings of the seventh ACM symposium on Solid modeling and applications
3-Dimensional Euclidean Voronoi Diagrams of Lines with a Fixed Number of Orientations
SIAM Journal on Computing
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
An exact and efficient approach for computing a cell in an arrangement of quadrics
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
On the computation of an arrangement of quadrics in 3D
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
Computational Geometry: Theory and Applications
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Thirty years of Polynomial System Solving, and now?
Journal of Symbolic Computation
Triangulations of line segment sets in the plane
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Journal of Symbolic Computation
Verified error bounds for real solutions of positive-dimensional polynomial systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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We give a complete description of the Voronoi diagram of three lines in R3. In particular, we show that the topology of the Voronoi diagram is invariant for three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. The trisector consists of four unbounded branches of either a non-singular quartic or of a cubic and line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. The proof technique, which relies heavily upon modern tools of computer algebra, is of interest in its own right. This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semi-algebraic tests for separating the two connected components of each two-dimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semi-algebraic tests.