Shape preserving least-squares approximation by polynomial parametric spline curves

doi:10.1016/S0167-8396(97)00004-6

Computer Aided Geometric Design

Volume 14, Issue 8, October 1997, Pages 731-747

https://doi.org/10.1016/S0167-8396(97)00004-6 Get rights and content

Abstract

This article presents a method for shape preserving least-squares approximation. This method generalizes an algorithm of Dierckx (1980, 1993) to the case of planar parametric curves. Using a reference curve we generate linear sufficient conditions for the convexity of the approximant. This leads us to a quadratic programming problem which can be solved exactly, e.g., with an active set strategy.

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In the isogeometric analysis framework, a computational domain is exactly described using the same representation as the one employed in the CAD process. For a CAD object, various computational domains can be constructed with the same shape but with different parameterizations; however one basic requirement is that the resulting parameterization should have no self-intersections. Moreover we will show, with an example of a 3D thermal conduction problem, that different parameterizations of a computational domain have different impacts on the simulation results and efficiency in isogeometric analysis. In this paper, a linear and easy-to-check sufficient condition for the injectivity of a trivariate B-spline parameterization is proposed. For problems with exact solutions, we will describe a shape optimization method to obtain an optimal parameterization of a computational domain. The proposed injective condition is used to check the injectivity of the initial trivariate B-spline parameterization constructed by discrete Coons volume method, which is a generalization of the discrete Coons patch method. Several examples and comparisons are presented to show the effectiveness of the proposed method. During the refinement step, the optimal parameterization can achieve the same accuracy as the initial parameterization but with less degrees of freedom.
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Shape preserving curve approximation is considered for an ordered set of spatial data points. Linear sufficient conditions for shape preserving criteria on convexity, inflection, collinearity and coplanarity are derived to ensure the approximant mimics the shape of the data. The cubic B-spline is used to generate the $C^{2}$ shape preserving approximation curve as this type of spline enables the shape preserving constraints to be formulated in a simple way.
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Boolean surfaces with shape constraints
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Citation Excerpt :
Despite its importance in practical applications, the shape preserving approximation has not received considerable attention. To the best of our knowledge, the problem of constructing shape preserving approximating planar or spatial curves has been addressed only in the papers [4–6,14]; similarly, [8] seems to be the only paper dealing with shape preserving parametric surfaces. In order to facilitate the comprehension of the following sections, we give here a brief account of the main steps which have driven our recent researches in this field.
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Citation Excerpt :
He exploits this fact in order to construct convex or monotone spline approximants. Jüttler [15] proposes the use of a reference curve in order to impose convexity constraints, i.e., given inflection points, to the approximating spline. Costantini et al. [3,4] use variable degree polynomial splines instead of exponential splines in order to get shape-preserving approximants for 2D and 3D parametric curves.
Tension can be applied to cubic splines in order to avoid undesired spurious oscillations. This leads to the well-known (exponential) spline in tension. It is crucial but unfortunately difficult to find suitable tension parameters of interpolating splines in tension. Instead of heuristics, we propose a simultaneous knot placing and tension setting algorithm for least-squares splines in tension which includes interpolating splines in tension as a special case. Moreover, the splines presented here are the foundation of exponential surface splines on fairly arbitrary meshes [K.O. Riedel, Two-dimensional splines on fairly arbitrary meshes, ZAMM—Z. Angew. Math. Mech. 85(3) (2005) 176–188].

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Shape preserving least-squares approximation by polynomial parametric spline curves

Abstract

Computer Aided Geometric Design

Quadratic Programming

Approximation with Aesthetic Constraints

An algorithm for cubic spline fitting with convexity constraints

Computing

Curve and Surface Fitting with Splines

An algorithm for computing constrained smoothing spline functions

Numer. Math.

Least squares data fitting using shape preserving piecewise approximations

Numer. Algorithms