Tor Dokken

Polynomial splines over locally refined box-partitions

We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled B-splines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using some recent dimension formulas.

Finding intersections of B-spline represented geometries using recursive subdivision techniques

Problems related to intersections are transformed to finding the zeroes of B-spline represented functions of one, two, three or four variables. The problems treated are intersections between B-spline represented geometries, implicitly represented geometries and B-spline represented geometries, extremal points on B-spline represented geometries, closest points between B-spline geometries and silhoutte curves on a B-spline surface. The B-spline represented geometries discussed are curves and surfaces. The zeroes of the functions are found by using recursive subdivision techniques.

Approximate implicitization using monoid curves and surfaces

Abstract This paper presents an approach to finding an approximate implicit equation and an approximate inversion map of a planar rational parametric curve or a rational parametric surface. High accuracy of the approximation is achieved with a relatively small number of low-degree curve segments or surface patches. By using monoid curves and surfaces, the method eliminates the undesirable singularities and “phantom” branches normally associated with implicit representation. The monoids are expressed in exact implicit and parametric equations simultaneously, and upper bounds are derived for the approximate errors of implicitization and inversion equations.

Approximate Implicitization Using Linear Algebra

In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating point implementation in computer aided geometric design (CAGD) systems. We unify the approaches under the names of commonly known polynomial basis functions, and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.

Locally refined spline surfaces for representation of terrain data

In this paper we describe the use of a novel representation, LR B-spline surfaces, and apply this representation in the treatment of geographical data. These data sets are typically very large and LR B-spline surfaces offer a compact representation of overall smooth data with local details. We briefly describe the properties of the LR B-spline representation, and discuss the details of two approximation methods adapted for LR B-splines: least squares approximation and multilevel B-spline approximation (MBA). The described techniques are demonstrated on several examples of terrain data in the form of point clouds. Graphical abstractAn LR B-spline surface approximating a point cloud representing Gaustatoppen in Norway. The point cloud covers 7kmi?7km and orth is pointing upwards to the right. The photograph has a slightly different view.Display Omitted HighlightsLR B-splines is a new surface representation for local refinement of spline spaces.Least squares approximation is adapted for LR B-splines.Multilevel B-spline approximation is adapted for LR B-splines.The methods are applied to approximate terrain data.

Spline surface intersections optimized for GPUs

A commodity-type graphics card with its graphics processing unit (GPU) is used to detect, compute and visualize the intersection of two spline surfaces, or the self-intersection of a single spline surface. The parallelism of the GPU facilitates fast and efficient subdivision and bounding box testing of smaller spline patches and their corresponding normal subpatches. This subdivision and testing is iterated until a prescribed level of accuracy is reached, after which results are returned to the main computer. We observe speedups up to 17 times relative to a contemporary 64 bit CPU.

Isogeometric Representation and Analysis - Bridging the Gap between CAD and Analysis

In 2005 Prof. Thomas J. R. Hughes proposed isogeometric analysis using volumetric NURBS elements rather than traditional finite elements for analysis. NURBS is the standard approach for representation of free form curves and sculptured surfaces in Computer Aided Design, and can represent elementary shapes such as sphere, cylinders, and torus exactly. Used in analysis NURBS consequently offers exact geometry representation, simplified design optimization and tighter integration of analysis and CAD. In this paper we address different aspects of isogeometric representation and analysis with a main focus on the relation to CAD, and how CAD can change to improve analysis by incorporating isogeometric representation.

Real-Time Algebraic Surface Visualization

We demonstrate a ray tracing type technique for rendering algebraic surfaces using programmable graphics hardware (GPUs). Our approach allows for real-time exploration and manipulation of arbitrary real algebraic surfaces, with no pre-processing step, except that of a possible change of polynomial basis.

An Introduction to General-Purpose Computing on Programmable Graphics Hardware

Using graphics hardware for general-purpose computations (GPGPU) has for selected applications shown a performance increase of more than one order of magnitude compared to traditional CPU implementations. The intent of this paper is to give an introduction to the use of graphics hardware as a computational resource. Understanding the architecture of graphics hardware is essential to comprehend GPGPU-programming. This paper first addresses the fixed functionality graphics pipeline, and then explains the architecture and programming model of programmable graphics hardware. As the CPU is instruction driven, while a graphics processing unit (GPU) is data stream driven, a good CPU algorithm is not necessarily well suited for GPU implementation. We will illustrate this with some commonly used GPU algorithms. The paper winds up with examples of GPGPU-research at SINTEF within simulation, visualization, image processing, and geometry processing.

Piecewise approximate implicitization: experiments using industrial data

1 Institute of Applied Geometry, Johannes Kepler University, Altenberger Str. 69, 4040 Linz, Austria; www.ag.jku.at, Email: firstname.lastname@jku.at 2 SINTEF ICT, P.O. Box 124 Blindern, N–O314 Oslo, Norway; www.math.sintef.no, Email: [Jan.B.Thomassen|Tor.Dokken]@sintef.no 3 Center of Mathematics for Applications, P.O.Box 1053 Blindern, N–0316 Oslo, Norway; www.cma.uio.no, Email: jan.b.thomassen@cma.uio.no