DETERMINATION OF EUCLIDEAN DISTANCES FOR SYMMETRY MOLECULES

Abstract This paper represents the geometric analysis of molecular surfaces of the molecules, indicates the blending operation of an atoms constitute to the small molecules. The decision which indicates advantages of Euclidean Voronoi diagram of an atom includes the blending surface among the atoms to make a fundamental study of docking, interactions with macromolecules. The algorithm which proposes the topological part of surfaces discussed through the Euclidean Voronoi Diagram of various accessibility procedures

Computing a Compact Spline Representation of the Medial Axis Transform of a 2D Shape

We present a full pipeline for computing the medial axis transform of an arbitrary 2D shape. The instability of the medial axis transform is overcome by a pruning algorithm guided by a user-defined Hausdorff distance threshold. The stable medial axis transform is then approximated by spline curves in 3D to produce a smooth and compact representation. These spline curves are computed by minimizing the approximation error between the input shape and the shape represented by the medial axis transform. Our results on various 2D shapes suggest that our method is practical and effective, and yields faithful and compact representations of medial axis transforms of 2D shapes.Comment: GMP14 (Geometric Modeling and Processing

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201

Exploring cavity dynamics in biomolecular systems

Background The internal cavities of proteins are dynamic structures and their dynamics may be associated with conformational changes which are required for the functioning of the protein. In order to study the dynamics of these internal protein cavities, appropriate tools are required that allow rapid identification of the cavities as well as assessment of their time-dependent structures. Results In this paper, we present such a tool and give results that illustrate the applicability for the analysis of molecular dynamics trajectories. Our algorithm consists of a pre-processing step where the structure of the cavity is computed from the Voronoi diagram of the van der Waals spheres based on coordinate sets from the molecular dynamics trajectory. The pre-processing step is followed by an interactive stage, where the user can compute, select and visualize the dynamic cavities. Importantly, the tool we discuss here allows the user to analyze the time-dependent changes of the components of the cavity structure. An overview of the cavity dynamics is derived by rendering the dynamic cavities in a single image that gives the cavity surface colored according to its time-dependent dynamics. Conclusion The Voronoi-based approach used here enables the user to perform accurate computations of the geometry of the internal cavities in biomolecules. For the first time, it is possible to compute dynamic molecular paths that have a user-defined minimum constriction size. To illustrate the usefulness of the tool for understanding protein dynamics, we probe the dynamic structure of internal cavities in the bacteriorhodopsin proton pump

Doctor of Philosophy

dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges

ARTIST-DRIVEN FRACTURING OF POLYHEDRAL SURFACE MESHES

This paper presents a robust and artist driven method for fracturing a surface polyhedral mesh via fracture maps. A fracture map is an undirected simple graph with nodes representing positions in UV-space and fracture lines along the surface of a mesh. Fracture maps allow artists to concisely and rapidly define, edit, and apply fracture patterns onto the surface of their mesh. The method projects a fracture map onto a polyhedral surface and splits its triangles accordingly. The polyhedral mesh is then segmented based on fracture lines to produce a set of independent surfaces called fracture components, containing the visible surface of each fractured mesh fragment. Subsequently, we utilize a Voronoi-based approximation of the input polyhedral mesh’s medial axis to derive a hidden surface for each fragment. The result is a new watertight polyhedral mesh representing the full fracture component. Results are aquired after a delay sufficiently brief for interactive design. As the size of the input mesh increases, the computation time has shown to grow linearly. A large mesh of 41,000 triangles requires approximately 3.4 seconds to perform a complete fracture of a complex pattern. For a wide variety of practices, the resulting fractures allows users to provide realistic feedback upon the application of extraneous forces