Interrogative visualization refers to the process of interactive computer graphics display and accurate quantitative querying of physical data. Quantitative querying includes search for metric, combinatorial and topological information. To support this paradigm, we build uniform, compact, co-registered representations (spline models) of multiple physical data fields over the same domain. Dense, unstructured volumetric scalar fields are approximated by C 1 -continuous trivariate polynomial spline functions of low degree. Additionally, these spline functions are also used to model scattered scalar fields sampled over a manifold surface in the volume. Using both implicit polynomial spline surfaces and trivariate polynomial spline functions allows for model representations of both manifold and associated scalar fields over the same spatial decomposition. Quantitative querying is made efficient by utilizing various search structures over the modeled field data. 1 Introduction I...

this report is as follows:

Automatic simplification of polyhedral objects is a major topic in many computer graphics applications. This work discusses simplification algorithms for the generation of a multiresolution family of solid representations from an initial polyhedral solid. We introduce the Discretized Polyhedra Simplification (DPS), a framework for polyhedra simplification using space decomposition models. The DPS is based on a new error measurement and provides a sound scheme for error-bounded, geometry and topology simplification while preserving the validity of the model. A method following this framework, Direct DPS, is presented and discussed. Direct DPS uses an octree for topology simplification and error control, and generates valid solid representations. Our method is also able to generate approximations which do not interpenetrate the original model, either being completely contained in the input solid or bounding it. Unlike most of the current methods, restricted to triangle meshes, our algorithm can deal and also produces faces with arbitrary complexity. An extension of the Direct method for appearance preservation, called Hybrid DPS, is also discussed.

An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic B'ezier curves meeting with tangent plane continuity at the vertices. This curve network is extended to a smooth surface by replacing each of the networks facets with a split patch consisting of three triangular B'ezier patches. The remaining degrees of freedom of the curve network and the split patches are determined by minimizing a quadratic functional. This optimization process works either for the curve network and the split patches separately or in one simultaneous step. The second variant of our algorithm is based on the construction of an optimized curve network with higher continuity. Examples demonstrate the quality of the different methods. 1 Introduction The reconstruction of a surface from a set of (a priori unorganized) points as well as the design of surfaces with a...

In this work we present the principles and applications of geometry simplification, focusing on simplification of polygonal representations of solids and surfaces. Related concepts such as multiresolution, level-of-detail and geometry compression are also discussed. A characterization of surface simplification methods is presented, including a classification, review and evaluation of the more relevant methods. 1 Contents 1 Introduction 5 2 Geometric modeling 6 2.1 Aims and branches of geometric modeling . . . . . . . . . . . . . . . . . . . . 6 2.2 Three-level view of modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Mathematical models of solids and surfaces . . . . . . . . . . . . . . . . . . . 7 2.4 Representation of solids and surfaces . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Representation of visual information . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Polyhedral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

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This report is an inventory of requirements in terms of geometric datastructures and algorithms that are needed in the four application areas. The list of requirements has been assembled in collaboration with of our industrial partners. These requirements are collected in order to see what functionality a useful computational geometry algorithms library, in particular cgal, could provide. However, not all functionality mentioned in this report should necessarily be part of the cgal library. Rather, we want that cgal

Automatic simplification of polyhedral objects is a major topic in many computer graphics applications. In this work, simplification algorithms for the generation of a multiresolution family of solid representations from an initial polyhedral solid are discussed. We introduce the Discretized Polyhedra Simplification (DPS), a framework for polyhedra simplification using space decomposition models. DPS is based on a new error distance and provides a sound scheme for error-bounded, geometry and topology simplification while preserving the validity of the model. A method following this framework, trihedral DPS, is presented and discussed. Trihedral DPS uses an octree for topology simplification and error control, and generates valid solid representations. Our method is also able to generate approximations that do not interpenetrate the original model, either being completely contained in the input solid or bounding it. Unlike most of the current methods, restricted to triangle meshes, our ...

Introduction Thus far in this book we have mostly encountered spline curves and surfaces whose parameter domains are subsets of the real line or the Euclidean plane. In particular, in several chapters of this book we got accustomed to the idea that a spline surface is the graph of a bivariate real-valued function or, alternatively, a parametric surface, which is the image of a planar domain under a vector function, or a collection of such functions. The parametric or free-form surfaces that are typically considered in the CAGD literature are composite surfaces consisting of a collection of individual surface patches of the form f i (S i ), where each of these corresponds to a three-component vector function f i : S i ! IR 3 ; i = 1; : : : ; N; (1) whose domain S i is a "simple" planar region, such as the standard triang

The problem of three-dimensional surface reconstruction from a set of unorganized set of points sampled from the surface of an object is important in many applications, such as computer vision, computer-aided design and solid modeling. In recent years several algorithms have been presented for surface reconstruction with provable guarantee on the closeness of the approximation. This thesis presents an algorithm with topological guarantee for reconstruction of a piecewise linear watertight approximation. In undersampled and non-smooth regions, where the sampling condition is not met for the theoretical guarantee, this method still constructs a good