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Showing papers by "Herbert Edelsbrunner published in 1996"


Journal ArticleDOI
TL;DR: If the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation and the algorithm takes expected time at mostO(nlogn+n[d/2]).
Abstract: A set ofn weighted points in general position in źd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.

341 citations


Journal ArticleDOI
TL;DR: A new approach to triangulating the surface of a molecule under the three models, which is fast, robust, and results in topologically correct triangulations is presented.

81 citations


Journal ArticleDOI
TL;DR: A tight Θ( n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to then given lines and an algorithm that tests the “towering property” inO(n2+ɛ) time.
Abstract: Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements ofn lines in three-dimensional space. Our main results include:1.A tight ź(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to then given lines.2.A similar bound of ź(n3) for the complexity of the set of all lines passing above then given lines.3.A preprocessing procedure usingO(n2+ź) time and storage, for anyź>0, that builds a structure supportingO(logn)-time queries for testing if a line lies above all the given lines.4.An algorithm that tests the "towering property" inO(n2+ź) time, for anyź>0; don given red lines lie all aboven given blue lines? The tools used to obtain these and other results include Plucker coordinates for lines in space andź-nets for various geometric range spaces.

67 citations


Proceedings Article
01 Jan 1996
TL;DR: Pockets can be efficiently constructed by an algorithm based on alpha complexes as discussed by the authors, which is implemented and applied to proteins with known three-dimensional conformations and can be used to identify identifiable regions in the complement space.
Abstract: The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm based on alpha complexes. The algorithm is implemented and applied to proteins with known three-dimensional conformations.

37 citations


Journal ArticleDOI
TL;DR: This work has developed general modeling software for a Cave Automatic Virtual Environment (CAVE), one of its applications is modeling 3D protein structures, generating both outside-in and inside-out views of geometric models.
Abstract: We have developed general modeling software for a Cave Automatic Virtual Environment (CAVE); one of its applications is modeling 3D protein structures, generating both outside-in and inside-out views of geometric models. An advantage of the CAVE over other virtual environments is that multiple viewers can observe the same scene at the same time and place. Our software is scalable-from high-end virtual environments such as the CAVE, to mid-range immersive desktop systems, down to low-end graphics workstations. In the current configuration, a parallel Silicon Graphics Power Challenge supercomputer architecture performs the computationally intensive construction of surface patches remotely, and sends the results through the I-WAY (Information Wide Area Year) using VBNS (Very-high-Bandwidth Network Systems) to the graphics machines that drive the CAVE and our graphics visualization software, Valvis (Virtual ALpha shapes VISualizer).

36 citations


01 Jan 1996
TL;DR: In this article, the Delaunay complex is maintained dynamically as a molecular shape changes over time using the theory of Alpha shapes, which is applicable to the analysis of molecular shape.
Abstract: Macromolecules such as proteins and DNA have complex spatial structures which are often important for their biological functions. The idea of molecular shape and shape complementarity play crucial roles in protein folding, conformational stability, molecular solubility, crystal packing, and docking. Various models based on a union of balls representation have been suggested for representing molecular shape, including the van der Waals model, the solvent accessible surface, and the molecular surface. In this thesis, we present several extensions to the theory of Alpha shapes applicable to the analysis of molecular shape. First, we define a ``pocket'''' of a molecule, which is intuitively a `depression'', `canyon'', or `cavity'' of a molecule. The definition is based on mathematical notions of relative distance. We give efficient algorithms for computing pockets and examples of their application. Secondly, we look at the issue of maintaining shape dynamically as a molecule changes over time. Topological analysis of the changing structure can yield information about the function of the molecule. We describe algorithms and their implementations for dynamically maintaining the Delaunay complex, the basis for shape analysis. These algorithms have been implemented, and experimental results are reported. Finally, we discuss techniques for modelling uniform growth of the atoms of a molecule. The solvent accessible and molecular surface models of a molecule are based on such growth, and the algorithms presented here efficiently compute these models for all probe sizes.

6 citations


Proceedings ArticleDOI
01 Jul 1996
TL;DR: A general approach to geometric modeling suitable for the Cave Automatic Virtual Environment is described, based on alpha complexes, and some of its capabilities are demonstrated by applying it to the study of biomolecules.
Abstract: Virtual environments open up new opportunities and challenges for geometric modeling systems. A general approach to geometric modeling suitable for the Cave Automatic Virtual Environment is described. The approach is based on alpha complexes, and some of its capabilities are demonstrated by applying it to the study of biomolecules.

5 citations


01 Jan 1996
TL;DR: A new approach to triangulating the surface of a molecule under the three models, which is fast, robust, and results in typologically correct triangulations is presented, which can be used to accurately compute the shape of cavities in proteins.
Abstract: Questions of chemical reactivity can often be cast as questions of molecular geometry. Common geometric models for proteins and other molecules are the space filling diagram, the solvent accessible surface and the molecular surface. In this thesis we present a new approach to triangulating the surface of a molecule under the three models, which is fast, robust, and results in typologically correct triangulations. Our algorithm can also be used to accurately compute the shape of cavities in proteins. Graphical display of molecular surfaces, typically defined by a large number of atoms modeled by overlapping spherical balls, requires the rendering of visually pleasing approximations of spheres and sphere patches. We present algorithms to construct piecewise linear approximations of spheres and sphere patches. In this context, we develop the notion of a constrained convex hull and present an algorithm for constructing it.

4 citations