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Kinetic Stable Delaunay Graphs
"... The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in R 2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay ..."
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Cited by 8 (3 self)
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The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in R 2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunay triangulation. SDG is defined in terms of a parameter α> 0, and consists of Delaunay edges pq for which the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least α. We prove several interesting properties of SDG and describe two kinetic data structures for maintaining it. Both structures use O ∗ (n) storage. They process O ∗ (n 2) events during the motion, each in O ∗ (1) time, provided that the points of P move along algebraic trajectories of bounded degree; the O ∗ (·) notation hides multiplicative factors that are polynomial in 1/α and polylogarithmic in n. The first structure is simpler but the dependency on 1/α in its performance is higher. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems—Geometrical problems and computations; G.2.1 [Discrete mathematics]: Combinatorics— Combinatorial algorithms
A Kinetic Triangulation Scheme for Moving Points in The Plane ∗
"... We present a simple randomized scheme for triangulating a set P of n points in the plane, and construct a kinetic data structure which maintains the triangulation as the points of P move continuously along piecewise algebraic trajectories of constant description complexity. Our triangulation scheme ..."
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We present a simple randomized scheme for triangulating a set P of n points in the plane, and construct a kinetic data structure which maintains the triangulation as the points of P move continuously along piecewise algebraic trajectories of constant description complexity. Our triangulation scheme experiences an expected number of O(n 2 βs+2(n) log 2 n) discrete changes, and handles them in a manner that satisfies all the standard requirements from a kinetic data structure: compactness, efficiency, locality and responsiveness. Here s is the maximum number of times where any specific triple of points of P can become collinear, βs+2(q) = λs+2(q)/q, and λs+2(q) is the maximum length of DavenportSchinzel sequences of order s + 2 on n symbols. Thus, compared to the previous solution of Agarwal et al. [4], we achieve a (slightly) improved bound on the number of discrete changes in the triangulation. In addition, we believe that our scheme is simpler to implement and analyze. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algorithms and problems—Geometrical problems and computations; G.2.1 [Discrete mathematics]: Combinatorics—Combinatorial algorithms
A Lagrangian Approach to Dynamic Interfaces through Kinetic Triangulation of the Ambient Space
 COMPUTER GRAPHICS FORUM
, 2007
"... In this paper, we propose a robust and efficient Lagrangian approach for modeling dynamic interfaces between different materials undergoing large deformations and topology changes, in two dimensions. Our work brings an interesting alternative to popular techniques such as the level set method and th ..."
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In this paper, we propose a robust and efficient Lagrangian approach for modeling dynamic interfaces between different materials undergoing large deformations and topology changes, in two dimensions. Our work brings an interesting alternative to popular techniques such as the level set method and the particle level set method, for twodimensional and axisymmetric simulations. The principle of our approach is to maintain a twodimensional triangulation which embeds the onedimensional polygonal description of the interfaces. Topology changes can then be detected as inversions of the faces of this triangulation. Each triangular face is labeled with the type of material it contains. The connectivity of the triangulation and the labels of the faces are updated consistently during deformation, within a neat framework developed in computational geometry: kinetic data structures. Thanks to the exact computation paradigm, the reliability of our algorithm, even in difficult situations such as shocks and topology changes, can be certified. We demonstrate the applicability and the efficiency of our approach with a series of numerical experiments in two dimensions. Finally, we discuss the feasibility of an extension to three dimensions.
Untangling Triangulations through Local Explorations
, 2007
"... In many applications it is often desirable to maintain a valid mesh within a certain domain that deforms over time. During a period for which the underlying mesh topology remains unchanged, the deformation moves the vertices of the mesh and thus potentially turns a mesh invalid, or as we call it, ta ..."
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In many applications it is often desirable to maintain a valid mesh within a certain domain that deforms over time. During a period for which the underlying mesh topology remains unchanged, the deformation moves the vertices of the mesh and thus potentially turns a mesh invalid, or as we call it, tangled. We introduce the notion of locally removable region, which is a certain tangled area in the mesh that allows for local removal and remeshing. We present an algorithm that is able to quickly compute, through local explorations, a minimal locally removable region containing a seed tangled region in the invalid mesh. By remeshing within this area, the seed tangled region can then be removed from the mesh without introducing any new tangled region. The algorithm is outputsensitive in the sense that it never explores outside the output region. Our algorithm exploits several novel insights into the structure of the tangled mesh, which may be of independent interest in contexts beyond mesh untangling.
On kinetic delaunay triangulations: A near quadratic bound for unit speed motions
 In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2013
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, 2013
"... The triangulation as an alternative painting medium ..."
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