A Method for Sparse-Matrix Computation of B-Spline Curves and Surfaces

Lakså, Arne

doi:10.1007/978-3-642-12535-5_95

Cited by 3 publications

(3 citation statements)

References 6 publications

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“…Further, it follows that between the knots t i and t i+1 can a polynomial B-spline curve be factorized as described in article [4]. In the following example, the factorizing of (4) is implemented on a 3 th degree B-spline curve,…”

Section: Factorization Of B-splinesmentioning

confidence: 99%

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

Lakså¹

2017

AIP Conference Proceedings

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Abstract. Interactive computer based geometry editing is very useful for designers and artists. Our goal has been to develop useful tools for geometry editing in a way that increases the ability for creative design.When we interactively editing geometry, we want to see the change happening gradually and smoothly on the screen. Preevaluation is a tool for increasing the speed of the graphics when doing interactive affine operation on control points and control surfaces. It is then possible to add details on surfaces, and change shape in a smooth and continuous way.We use pre-evaluation on basis functions, on blending functions and on local surfaces. Pre-evaluation can be made hierarchically and is thus useful for local refinements. Sampling and plotting of curves, surfaces and volumes can today be handled by the GPU and it is therefore important to have a structured organization and updating system to be able to make interactive editing as smooth and user friendly as possible. In the following, we will show a structure for pre-evaluation and an optimal organisation of the computation and we will show the effect of implementing both of these techniques.

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Section: Factorization Of B-splinesmentioning

confidence: 99%

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

Lakså¹

2017

AIP Conference Proceedings

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“…are intermediate points of de Casteljau's algorithm. This recursive convex linear interpolation between two points can be expressed in matrix form as outlined in [8]. As an example, computing from right to left, for n = 3:…”

Section: Univariate Bernstein Polynomials and Curves On Bézier Formmentioning

confidence: 99%

“…Such matrices are related to de Casteljau's corner cutting algorithm [7]. The univariate case of Bernstein factor matrices was addressed in [8], however, univariate Bernstein factor matrices seem to have been exposed and used prior to that, see e.g. [9], where the de Casteljau algorithm is expressed on matrix form, or the method presented in [10, algorithm 12].…”

Section: Introductionmentioning

confidence: 99%

Matrix Factorization of Multivariate Bernstein Polynomials

Dalmo¹

2015

Int. J. of Pure and Appl. Math.

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Ordinary univariate Bernstein polynomials can be represented in matrix form using factor matrices. In this paper we present the definition and basic properties of such factor matrices extended from the univariate case to the general case of arbitrary number of variables by using barycentric coordinates in the hyper-simplices of respective dimension. The main results in the paper are related to the design of an iterative algorithm for fast convex computation of multivariate Bernstein polynomials based on sparse-matrix factorization. In the process of derivation of this algorithm, we investigate some properties of the factorization, including symmetry, commutativity and differentiability of the factor matrices, and address the relevance of this factorization to the de Casteljau algorithm for evaluating curves and surfaces on Bézier form. A set of representative examples is provided, including a geometric interpretation of the de Casteljau algorithm, and representation by factor matrices of multivariate surfaces and their derivatives in Bézier form. Another new result is the observation that inverting the order of steps of a part of the new factorization algorithm provides a new, matrix-based, algebraic representation of a multivariate generalization of a special case of the de Boor-Cox computational algorithm.

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Non polynomial B-splines

Lakså¹

2015

AIP Conference Proceedings

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A Method for Sparse-Matrix Computation of B-Spline Curves and Surfaces

Cited by 3 publications

References 6 publications

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

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

Matrix Factorization of Multivariate Bernstein Polynomials

Non polynomial B-splines

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