Design of the CGAL 3D Spherical Kernel and application to arrangements of circles on a sphere

Castro, Pedro Machado Manhães de; Cazals, Frédéric; Loriot, Sébastien; Teillaud, Monique

doi:10.1016/j.comgeo.2008.10.003

Cited by 13 publications

(9 citation statements)

References 37 publications

(36 reference statements)

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“…Second, since a boundary point is found at the intersection of three input spheres, its coordinates are degree two algebraic numbers. We therefore store these points using the CGAL spherical kernel Spher-ical_kernel_3 [CCLT09], instantiated with K. The two options for K, referred to as the inexact and the exact kernels in the sequel, are:…”

Section: B1 Inner Approximationmentioning

confidence: 99%

Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse‐Graining

Cazals¹,

Dreyfus²,

Sachdeva

et al. 2014

Computer Graphics Forum

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International audienceChoosing balls which best approximate a 3D object is a non trivial problem. To answer it, we first address the inner approximation problem, which consists of approximating an object FO defined by a union of n balls with k < n balls defining a region FS ⊂ FO. This solution is further used to construct an outer approximation enclosing the initial shape, and an interpolated approximation sandwiched between the inner and outer approximations. The inner approximation problem is reduced to a geometric generalization of weighted max k-cover, solved with the greedy strategy which achieves the classical 1 − 1/e lower bound. The outer approximation is reduced to exploiting the partition of the boundary of FO by the Apollonius Voronoi diagram of the balls defining the inner approximation. Implementation-wise, we present robust software incorporating the calculation of the exact Delau-nay triangulation of points with degree two algebraic coordinates, of the exact medial axis of a union of balls, and of a certified estimate of the volume of a union of balls. Application-wise, we exhibit accurate coarse-grain molecular models using a number of balls 20 times smaller than the number of atoms, a key requirement to simulate crowded cellular environments

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Section: B1 Inner Approximationmentioning

confidence: 99%

Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse‐Graining

Cazals¹,

Dreyfus²,

Sachdeva

et al. 2014

Computer Graphics Forum

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“…First, we are not aware of any robust implementation to report the volume of a union of balls. A contrario, robust and optimized algorithms exist to handle surface arrangements [14], [15], [16]…”

Section: Application To Conformer Selectionsmentioning

confidence: 99%

“…The naive implementation was carried out using the Delaunay_3 and Alpha_shape_3 packages of the Computational Geometry Algorithms Library [26]. For the priority based version, we used the surface arrangements package described in [14], [15], [16], which is the only one, to the best of out knowledge, able to compute effectively the exact arrangement of circles on a sphere. In both cases, robustness issues are critical due to the density of conformers manipulated.…”

Section: The Priority-based Version Of Algorithmmentioning

confidence: 99%

On the Characterization and Selection of Diverse Conformational Ensembles with Applications to Flexible Docking

Loriot

Sachdeva

Bastard

et al. 2011

IEEE/ACM Trans. Comput. Biol. and Bioinf.

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To address challenging flexible docking problems, a number of docking algorithms pregenerate large collections of candidate conformers. To remove the redundancy from such ensembles, a central problem in this context is to report a selection of conformers maximizing some geometric diversity criterion. We make three contributions to this problem. First, we resort to geometric optimization so as to report selections maximizing the molecular volume or molecular surface area (MSA) of the selection. Greedy strategies are developed, together with approximation bounds. Second, to assess the efficacy of our algorithms, we investigate two conformer ensembles corresponding to a flexible loop of four protein complexes. By focusing on the MSA of the selection, we show that our strategy matches the MSA of standard selection methods, but resorting to a number of conformers between one and two orders of magnitude smaller. This observation is qualitatively explained using the Betti numbers of the union of balls of the selection. Finally, we replace the conformer selection problem in the context of multiple-copy flexible docking. On the aforementioned systems, we show that using the loops selected by our strategy can improve the result of the docking process.

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“…The half-edge data structure encodes the boundary of the union of balls, as computed in [19]. A certified embedding in 3D is obtained thanks to the robust geometric operations described in [20].…”

Section: Shelling Protein Binding Patches: Detailed Algorithmmentioning

confidence: 99%

Shape Matching by Localized Calculations of Quasi-Isometric Subsets, with Applications to the Comparison of Protein Binding Patches

Cazals

Malod-Dognin

2011

Pattern Recognition in Bioinformatics

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Abstract. Given a protein complex involving two partners, the receptor and the ligand, this paper addresses the problem of comparing their binding patches, i.e. the sets of atoms accounting for their interaction. This problem has been classically addressed by searching quasi-isometric subsets of atoms within the patches, a task equivalent to a maximum clique problem, a NP-hard problem, so that practical binding patches involving up to 300 atoms cannot be handled.We extend previous work in two directions. First, we present a generic encoding of shapes represented as cell complexes. We partition a shape into concentric shells, based on the shelling order of the cells of the complex. The shelling order yields a shelling tree encoding the geometry and the topology of the shape. Second, for the particular case of cell complexes representing protein binding patches, we present three novel shape comparison algorithms. These algorithms combine a Tree Edit Distance calculation (TED) on shelling trees, together with Edit operations respectively favoring a topological or a geometric comparison of the patches. We show in particular that the geometric TED calculation strikes a balance, in terms of accuracy and running time, between purely geometric and topological comparisons, and we briefly comment on the biological findings reported in a companion paper.

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Design of the CGAL 3D Spherical Kernel and application to arrangements of circles on a sphere

Cited by 13 publications

References 37 publications

Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse‐Graining

Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse‐Graining

On the Characterization and Selection of Diverse Conformational Ensembles with Applications to Flexible Docking

Shape Matching by Localized Calculations of Quasi-Isometric Subsets, with Applications to the Comparison of Protein Binding Patches

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