Arne Lakså

A Model of Enterprise Integration and Collaboration Tools and Communication Infrastructure for Inter- Enterprise Collaboration

This paper first discusses a model of enterprise integration that consists of a micro level, a midcro level, and a macro level integration. Secondly, the architecture for midcro inter-enterprise integration is composed consisting of three layers, a communication layer, a collaboration layer, and an application layer. To the collaboration layer, synchronous and asynchronous collaborations are discussed and a toolkit for those collaborations is studied and assembled. Then to the communication layer, a communication infrastructure for a research test-bed and a project for midcro inter-enterprise integration is described.

First Instances of Generalized Expo‐Rational Finite Elements on Triangulations

In this communication we consider a construction of simplicial finite elements on triangulated two‐dimensional polygonal domains. This construction is, in some sense, dual to the construction of generalized expo‐rational B‐splines (GERBS). The main result is in the obtaining of new polynomial simplicial patches of the first several lowest possible total polynomial degrees which exhibit Hermite interpolatory properties. The derivation of these results is based on the theory of piecewise polynomial GERBS called Euler Beta‐function B‐splines. We also provide 3‐dimensional visualization of the graphs of the new polynomial simplicial patches and their control polygons.

A method for sparse-matrix computation of b-spline curves and surfaces

Matrix methods of computing B-spline curves and surfaces have been considered in the work of several authors Here we propose a new, more general matrix formulation and respective upgraded notation The new approach is based on non-commutative operator splitting, where the domain and range of every factoring operator differ by one dimension, and the factoring operators are represented by a product of sparse rectangular matrices with expanding dimensions differing by 1, so that these matrices are d×(d+1)-dimensional (with d increasing with an increment of 1) and have nonzero values only on their two main diagonals (ai,i) and (ai,i+1), i=1,...,d In this new matrix formulation it is possible to obtain the generation of the B-spline basis and the algorithms of de Casteljau and Cox–de Boor in a very lucid unified form, based on a single matrix product formula This matrix formula also provides an intuitively clear and straightforward unified approach to corner cutting, degree elevation, knot insertion, computing derivatives and integrals in matrix form, interpolation, and so on For example, computing the matrix product in the formula from left to right results in the successive iterations of the de Casteljau algorithm, while computing it from right to left is equivalent to the successive iterations in the Cox–de Boor algorithm Although the new matrix factorization is essentially non-commutative, in Theorem 1 we formulate and prove an important commutativity relation between this matrix factorization and the operator of differentiation We use this relation further to propose a new, considerably more concise form of matrix notation for B-splines, with respective efficient computation based on sparse-matrix multiplication.

Non polynomial B-splines

B-splines are the de facto industrial standard for surface modelling in Computer Aided design. It is comparable to bend flexible rods of wood or metal. A flexible rod minimize the energy when bending, a third degree polynomial spline curve minimize the second derivatives. B-spline is a nice way of representing polynomial splines, it connect polynomial splines to corner cutting techniques, which induces many nice and useful properties. However, the B-spline representation can be expanded to something we can call general B-splines, i.e. both polynomial and non-polynomial splines. We will show how this expansion can be done, and the properties it induces, and examples of non-polynomial B-spline.

Smooth partition of unity with Hermite interpolation: applications to image processing

We explore the one-to one correspondence between parametric surfaces in 3D and two dimensional color images in the RGB color space. For the case of parametric surfaces defined on general parametric domains recently a new approximate geometric representation has been introduced1 which also works for manifolds in higher dimensions. This new representation has a form which is a generalization to the B´ezier representation of parametric curves and tensorproduct surfaces. The main purpose of the paper is to discuss how the so generated technique for modeling parametric surfaces can be used for respective modification (re-modeling) of images. We briefly consider also some of the possible applications of this technique.

Blending functions for hermite interpolation by beta-function b-splines on triangulations

In the present paper we compute for the first time Beta-function B-splines (BFBS) achieving Hermite interpolation up to third partial derivatives at the vertices of the triangulation. We consider examples of BFBS with uniform and variable order of the Hermite interpolation at the vertices of the triangulation, for possibly non-convex star-1 neighbourhoods of these vertices. We also discuss the conversion of the local functions from Taylor monomial bases to appropriately shifted and scaled Bernstein bases, thereby converting the Hermite interpolatory form of the linear combination of BFBS to a new, Bezier-type, form. This conversion is fully parallelized with respect to the vertices of the triangulation and, for Hermite interpolation of uniform order, the load of the computations for each vertex of the computation is readily balanced.

Multivariate Hermite interpolation on scattered point sets using tensor‐product expo‐rational B‐splines

At the Seventh International Conference on Mathematical Methods for Curves and Surfaces, To/nsberg, Norway, in 2008, several new constructions for Hermite interpolation on scattered point sets in domains in Rn,n∈N, combined with smooth convex partition of unity for several general types of partitions of these domains were proposed in [1]. All of these constructions were based on a new type of B‐splines, proposed by some of the authors several years earlier: expo‐rational B‐splines (ERBS) [3].In the present communication we shall provide more details about one of these constructions: the one for the most general class of domain partitions considered.This construction is based on the use of two separate families of basis functions: one which has all the necessary Hermite interpolation properties, and another which has the necessary properties of a smooth convex partition of unity. The constructions of both of these two bases are well‐known; the new part of the construction is the combined use of these bases...

Pre-evaluation and interactive editing of B-spline and GERBS curves and surfaces

Manuscript version OA on free e-print servers / repositories. Publishers copyright and source policy: https://publishing.aip.org/authors/web-posting-guidelines Link to publishers version: https://doi.org/10.1063/1.5013999

Surface Constructions on Irregular Grids

“Big” surfaces defined on domains that can not be modeled on a single regular grid is typically made by joining several surfaces together with the aid of fillet surfaces or by intersecting the surfaces and joining them after trimming.

Surfaces from Curves on Triangular Surfaces in Barycentric Coordinates

Barycentric coordinates are coordinates in which a position is provided by a blending of a weighted point set where the weights sum up to 1. Bezier-triangles and ERBS-triangles are typical examples of use of Barycentric coordinates.