J.-D. Boissonant and M. Yvinec, Algorithmic Geometry, Cambridge University Press, Cambridge, UK, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The main theoretical result of this paper is stated in the following theorem: Theorem 2. The running time of Algorithm FRACTIONALTREE with CONTRACTPATH is in O(n log n) Proof. Sketch) To find Vq, we need to compute the maximum of m linear functions, which can be done in time O(m log m) see [2] for a proof) The resulting piecewise linear function has at most m breakpoints. In every iteration there is a number of breakpoints from CONTRACTPATH and a number of leaves with corresponding root values to be considered. We use binary search in each iteration to find a new interval (l, in) ....

J. D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.

....questions. I am still surprised of how fast we encountered opened problems of mathematics, such as the Poincare s conjecture. But due to the existence of those computational problems we also found some answers. 13 Introduction 1. 1 Motivations and applications Computational geometry [Boi98] has led to major improvements in computer graphics, robotics, and computer aided design. This field focuses mainly on discrete problems involving point sets, polygons, and polyhedrons, and uses combinatorial techniques to solve them, with emphasis on provable correctness, e#ciency, and ....

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.

....so that every segment joining two consecutive vertices sampled along a feature backbone is added to the list of constrained edges (see Figure 9) Figure 9: Constrained Delaunay triangulation. 4. 4 Construction With a Delaunay triangulation, we can already deduce an initial Voronoi Tessellation [6]. From this, we aim at building a weighted centroidal Voronoi tessellation [16] to improve the initial sampling obtained by error diffusion. Definition A weighted centroidal Voronoi tessellation is a Voronoi diagram such that the associated sites coincide with the center of mass of the ....

BOISSONNAT, J.-D., AND YVINEC, M. Algorithmic Geometry. Cambridge University Press, UK, 1998.

....Is this still true for more than two dimensions The answer is no in one direction, because for a change in class distributions, there are infinite many corresponding cost matrices due to many more degrees of freedom. The best algorithm for the convex hull generation is O(N log N Nd 2) 21][5]. In the 2 d case, it is relatively straightforward how to detect the trivial classifiers and the points for the minimum and maximum cases. However, not only there are computational limitations but representational ones. ROC analysis in two dimensions has a very nice and understandable ....

Boissonat, J.D.; Yvinec, M. Algorithmic Geometry. Cambridge University Press, 1998.

....unit square a = percent covered (c) on the extended domain Fig. 6. The weights of the quadrature points can be seen as the area of a zone of influence. To detect if the original surface is actually occluding an interaction, we compute the trapezoidal map of an arrangement of line segments [4, 8] over the segments of the contour of the original surface (see figure 5) For each trapeze, we store its status whether it is inside or outside of the original surface. Using randomized algorithms, trapezoidal maps can be constructed in time O(n log n) where n is the number of vertices. Once ....

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.

.... the case: one can compute such a face in O( t 2 (n) log n) randomized expected time, see [dBDS95, CEG 93] and Chapter 9, and in slightly worse, but still near linear deterministic time [GSS89] A fundamental tool in manipulating arrangements is the randomized incremental construction approach [BY98, Mul94] Here one computes a (sub) structure in an arrangement (e.g. a face in an arrangement of arcs) incrementally, by inserting the arcs (or surfaces) one by one, and by updating and maintaining the structure after each insertion. As is the case in most applications, reasonable (expected) ....

.... of the fundamental problems in computational geometry, and has received a lot of attention in recent years [SA95] As already mentioned, one of the basic techniques used for such constructions is based on randomized incremental construction of the vertical decomposition of the arrangement (see [BY98] for an example) If we are interested in only computing parts of the arrangement (e.g. a single face or a 7.2 Taking a Walk in a Planar Arrangement 95 to trim parts of the plane as the algorithm advances, so that it will not waste resources on regions which are no longer relevant. In ....

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J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, 1998. Translated by H. Bronnimann.

....is defined similarly. Each leaf node l contains a list of objects S l that intersect R l . Definition 17 kD tree. A kD tree in R d is BSP tree with all partitioning planes aligned with one of the coordinate axis a (0 a d) A precise definition of the polyhedron is more complex. See [20] for an example. 4 3 Visibility Along Line The problems of visibility along a single line seem rather simple since all visual events that may occur, are restricted to the given line. Many modern methods of image synthesis form the final image by sampling visibility between two or a chain of ....

....visual events [71] events involving an edge an vertex (VE) and events involving three edges (EEE) of some polygons. The VE events correspond to planes, the EEE events in general form quadratic surfaces. 37 We use plucker coordinates [131, 210] to map lines to projective five dimensional space [20]. The major advantage of this mapping is that it allows us to describe all visual events by means hyperplanes in plucker coordinates. This property greatly simplifies the approach since no explicit treatment of three dimensional quadratic surfaces is needed. In this section we introduce the the ....

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. cambridge University Press, 1998.

....problems involving objects like points, lines, polygons, and polyhedra. Over the past twenty years, the eld has developed a rich collection of solutions to a huge variety of geometric problems including intersection problems, visibility problems, and proximity problems. See the textbooks [15, 23, 18, 21, 6, 1] and the handbook [10] for an overview. The standard approach taken in computational geometry is the development of provably good and ecient solutions to problems. However, implementing these algorithms is not easy. The most common problems are the dissimilarity between fast oating point ....

....cause runtime overhead. In the sequel we give examples of the use of the generic programming paradigm in cgal. 2 Generic Programming in Geometric Computing One of the hallmarks of geometry is the use of transformations. Indeed, geometric transformations link several geometric structures together [6, 1]. For example, duality relates the problem of computing the intersection of halfplanes containing the origin to that of computing the convex hull of their dual points. The Voronoi diagram of a set of points is also dually related to its Delaunay triangulation, and this triangulation can be ....

[Article contains additional citation context not shown here]

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, 1998. translated by H. Bronnimann.

.... arcs of S, is one of the fundamental problems in computational geometry, and has received a lot of attention in recent years [SA95] One of the basic techniques used for such constructions is based on randomized incremental construction of the vertical decomposition of the arrangement (see [Mul94, BY98] Some applications of planar arrangements require the construction of only parts of the arrangement (e.g. a single face or a zone of some curve) and, usually, those parts have smaller This work has been supported by a grant from the U.S. Israeli Binational Science Foundation. Work by Micha ....

....obtained by erecting two vertical segments up and down from each vertex of A(S) i.e. each point of intersection between a pair of arcs and each endpoint of an arc) and by extending each of them until it either reaches an arc of S, or otherwise all the way to infinity. See, e.g. BY98, SA95] for more details concerning vertical decompositions. To simplify (though slightly abuse) the notation, we refer to the cells of A VD (S) as trapezoids. Computing the decomposed arrangement A VD (S) can be done as follows. Pick a random permutation hSi = hs 1 ; s n i of S. Put S i ....

[Article contains additional citation context not shown here]

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, 1998. Translated by H. Bronnimann.

.... by the arcs of S, is one of the fundamental problems in computational geometry, and has received a lot of attention in recent years [SA95] One of the basic techniques used for such problems is based on randomized incremental construction of the vertical decomposition of the arrangement (see [BY98] for an example) If we are interested in only computing parts of the arrangement (e.g. a single face or a zone) the randomized incremental technique can still be used, but it requires non trivial modifications [CEG 93, dBDS95] Intuitively, the added complexity is caused by the need to ....

....vertical pseudo trapezoids, obtained by erecting two vertical segments up and down from each vertex of A( S) i.e. points of intersections between pairs of arcs and endpoints of arcs) and extending each of them until it either reaches an arc of S, or otherwise all the way to infinity. See [BY98] for more details concerning vertical decomposition. To simplify (though slightly abuse) the notation, we refer to the cells of A VD ( S) as trapezoids. 2 A selection R of S is an ordered sequence of distinct elements of S. By a slight abuse of notation, we also denote by R the unordered ....

[Article contains additional citation context not shown here]

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, 1998. translated by H. Bronnimann.

.... we have implemented are: i) a naive algorithm goes over all the edges in the map to nd the location of the query point; ii) an ecient algorithm (the default one) which is based on the randomized incremental construction of a search structure through the vertical decomposition of the map [15], 62] 72] and (iii) a walk along a line (walk, for short) algorithm that is an improvement over the naive approach: nds the point s location by walking along a vertical line from in nity towards the query point. We remind the reader that our point location implementation handles general ....

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, 1998.

....2 Preliminaries Let R 2 be the Euclidean plane. In this paper, we denote by ha; bi the line passing through the points a and b [2] Let m be a point of R 2 . Let S (a; r) be a circle of centre a and radius r. Its equation is given by S (x) 0 where S (x) d (a; x) 2 Gamma r 2 (see [4]) The power of the point m with respect to the circle S is defined as [4] S (m) d (a; m) 2 Gamma r 2 . If S (m) 0, the point m is on S (a; r) if S (m) 0, the point m is inside S (a; r) if S (m) 0, m is outside S (a; r) If we take any line l passing through m and intersecting S ....

....denote by ha; bi the line passing through the points a and b [2] Let m be a point of R 2 . Let S (a; r) be a circle of centre a and radius r. Its equation is given by S (x) 0 where S (x) d (a; x) 2 Gamma r 2 (see [4] The power of the point m with respect to the circle S is defined as [4]: S (m) d (a; m) 2 Gamma r 2 . If S (m) 0, the point m is on S (a; r) if S (m) 0, the point m is inside S (a; r) if S (m) 0, m is outside S (a; r) If we take any line l passing through m and intersecting S (a; r) at the intersection points t and t 0 , we have the following ....

[Article contains additional citation context not shown here]

J-D. Boissonat and M. Yvinec, Algorithmic Geometry, Cambridge University Press, 1998.

....geometric problems involving objects like points, lines, polygons, and polyhedra. Over the past twenty years, the eld has developed a rich collection of solutions to a huge variety of geometric problems including intersection problems, visibility problems, and proximity problems. See the textbooks [15, 23, 18, 21, 6, 1] and the handbook [10] for an overview. The standard approach taken in computational geometry is the development of provably good and eOEcient solutions to problems. However, implementing these This work is partially supported by the ESPRIT IV LTR Projects No. 21957 (CGAL) and 28155 (GALIA) ....

....it doesn t cause runtime overhead. In the sequel we give examples of the use of the generic programming paradigm in Cgal. 2 Generic Programming in Geometric Computing One of the hallmarks of geometry is transformations. Indeed, geometric transformations link several geometric structures together [6, 1]. For example, duality relates the problem of computing the intersection of halfplanes containing the origin to that of computing the convex hull of their dual points. The Voronoi diagram of a set of points is also dually related to its Delaunay triangulation, and this triangulation can be ....

[Article contains additional citation context not shown here]

J.-D. Boissonnat and M. Yvinec. Algorithmic geometry. Cambridge University Press, UK, 1998. translated by H. Br#nnimann.

....point s location by walking along a line from infinity towards the query point. In the following sections we describe these algorithms and their implementation. 4. 1 Fast Point Location As mentioned above the default point location is based on Mulmuley s randomized, fully dynamic algorithm (see [1, 10]) We remind the reader that our point location implementation handles general finite planar maps. The subdivision is not necessarily monotone (each face boundary is a union of x monotone chains) nor connected and possibly contains holes. In addition the input may be x degenerate. Algorithm Our ....

Boissonnat, J.-D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press (1998)

....by the many applications in which geometric problems and algorithms play a fundamental role. The standard approach taken in computational geometry is the development of exact, provably correct and e cient solutions to geometric problems. See for example the text books [Mul93] O R94] dBvKOS97] BY98] and the handbook [GO97] The impact of computational geometry on application domains was minor up to a few years ago. On one hand, the research community has been developing more interest in application problems and real world conditions, and develops more software implementations of the most ....

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.

....on representation issues and especially insist on how the cases of degenerate dimensions are treated. The algorithmic issues are not examined in this short paper. 1 Introduction A three dimensional triangulation is a three dimensional simplicial complex, pure connected and without singularities [BY98] It is a set of cells (tetrahedra) such that two cells either do not intersect or share a common facet, edge or vertex. Generalizing the storage of 2D triangulations [tri99] to the 3D case, we choose to explicitly represent only cells and vertices, together with adjacency and incidence ....

J.-D. Boissonnat and M. Yvinec. Algorithmic geometry. Cambridge University Press, UK, 1998.

....objects like points, lines, polygons, and polyhedra. The eld has, over the past twenty years, developed a rich collection of solutions to a huge variety of geometric problems including intersection problems, visibility problems, and proximity problems. See textbooks [PS85, Mul93, O R94, dBvKOS97, BY98] or a handbook [GO97] for an overview. The standard approach taken in computational geometry is the development of exact, provably good and e cient solutions to problems. However, implementing these algorithms isn t easy. As a result, many useful geometric algorithms haven t found their way into ....

J. D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.

....problems involving objects like points, lines, polygons, and polyhedra. Over the past twenty years, the field has developed a rich collection of solutions to a huge variety of geometric problems including intersection problems, visibility problems, and proximity problems. See the textbooks [15, 23, 18, 21, 6, 1] and the handbook [10] for an overview. The standard approach taken in computational geometry is the development of provably good and efficient solutions to problems. However, implementing these algorithms is not easy. The most common problems are the dissimilarity between fast floating point ....

....cause runtime overhead. In the sequel we give examples of the use of the generic programming paradigm in cgal. 2 Generic Programming in Geometric Computing One of the hallmarks of geometry is the use of transformations. Indeed, geometric transformations link several geometric structures together [6, 1]. For example, duality relates the problem of computing the intersection of halfplanes containing the origin to that of computing the convex hull of their dual points. The Voronoi diagram of a set of points is also dually related to its Delaunay triangulation, and this triangulation can be ....

[Article contains additional citation context not shown here]

J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, 1998. translated by H. Bronnimann.

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Boissonnat, Jean-Daniel, and Yvinec, Mariette, Algorithmic Geometry, Cambridge University Press, UK, 1998.

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J.-D. Boissonant and M. Yvinec, Algorithmic Geometry, Cambridge University Press, Cambridge, UK, 1998.

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J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.

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J. D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, 1998.

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J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, 1998.

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J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, Cambridge, 1998.

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J.-D. BOISSONNAT AND M. YVINEC, Algorithmic Geometry, Cambridge University Press (1998).

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