Higher order Bézier circles

Chou, Jin J.

doi:10.1016/0010-4485(95)91140-g

Cited by 26 publications

(6 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The test for P 2 complexity outlined in Equation (55) will yield the P 2 complexity of the parameterization, which is not necessarily equal to the P 2 complexity of the geometry. For example, consider that it is possible to parametrize a circle by a collection of C 0 -continuous quadratic Bézier curves, or by a single C 1 -continuous quartic rational Bézier curve [9]. In both cases, the P 2 complexity of the geometry is P 2 = 2.…”

Section: Geometric Polynomial Complexitymentioning

confidence: 99%

Isogeometric unstructured tetrahedral and mixed-element Bernstein–Bézier discretizations

Engvall

Evans

2017

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

A well-known problem in the field of isogeometric analysis is that of surface-tovolume parameterization. In CAD packages, solid objects are represented by a collection of NURBS or T-spline surfaces, but to perform engineering analysis for many real world problems, we must find a way to parameterize the volumes of these objects as well. This has proven to be difficult using traditional isogeometric methods, as the tensor-product nature of trivariate NURBS and T-splines limit their ability to provide analysis suitable parameterizations of arbitrarily complex volumes. To overcome the limitations of trivariate NURBS and T-splines, we propose the use of unstructured Bernstein-Bézier discretizations. In earlier work, we demonstrated the feasibility of this approach in two dimensions through the construction of an automatic mesh generation environment capable of generating geometrically exact unstructured meshes comprised of rational Bézier triangles. This paper extends the concepts presented in our previous work to unstructured tetrahedral and mixed-element meshes in three dimensions. The main contributions of this paper are threefold. First, we present a framework for creating geometrically exact meshes comprised of rational Bézier hexahedra, tetrahedra, wedges and pyramids through degree elevation of suitable linear meshes. We additionally discuss how our approach may be applied to higher-order mesh generation for more traditional finite element approaches. Next, we propose two quality metrics for threedimensional curvilinear meshes, and discuss some important considerations for mesh quality of higher-order meshes. Finally, we demonstrate the analysis suitability of the meshes constructed using our framework through numerical examples.

show abstract

Section: Geometric Polynomial Complexitymentioning

confidence: 99%

Isogeometric unstructured tetrahedral and mixed-element Bernstein–Bézier discretizations

Engvall

Evans

2017

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

“…However, there exist other chord-length Bezier representations, namely those of higher degree derived from the standard one by inserting base points. For the cubic and quartic cases, such representations are subsumed in the formulae given by Chou (1995).…”

Section: Rational Representations Of Higher Degreementioning

confidence: 99%

Curves with rational chord-length parametrization

Sánchez-Reyes

Fernández–Jambrina

2008

Computer Aided Geometric Design

View full text Add to dashboard Cite

It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled.Keywords: Bezier circle; Bipolar coordinates; Chord-length parametrization; Equilateral hyperbola; Lemniscate of Bernoulli Chord-length parametrizationGiven a parametric curve p(t) over a certain domain t e [a,b], its chord-length at a given point p(7) is defined as (Farm, 2001(Farm, , 2006 Chord-length parametrization admits an intuitive physical interpretation (Fig. 1). Suppose you have a rubber band of unit length, on which we draw a unit scale representing the unit domain. Then we attach the rubber band to two fixed points A, B and trace a curve by holding the band tout, pressing on it using a pencil without friction. As the band is submitted to a common tension all along its length, it stretches linearly, and hence condition (1) is satisfied. In fact, we could draw any curve using this method, but in general the curve would not be parametrized by a rational function of the chord u. The question to solve is precisely finding the curves for which the resulting parametrization is rational

show abstract

“…Some researched around the use of Hermite polynomials [2,3,7]. Others focused on the Bezier formulation [13,17,9,6,8]. Piegl and Tiller [11] proposed a method for circular arc approximation with non-rational B-splines, it allowed the selection of any specified degree and the continuity level.…”

Section: Introductionmentioning

confidence: 99%

Approximation of Circular Arcs by C<sup>2</sup> Cubic Polynomial B-splines

2007

2007 10th IEEE International Conference on Computer-Aided Design and Computer Graphics

View full text Add to dashboard Cite

show abstract

Higher order Bézier circles

Cited by 26 publications

References 4 publications

Isogeometric unstructured tetrahedral and mixed-element Bernstein–Bézier discretizations

Isogeometric unstructured tetrahedral and mixed-element Bernstein–Bézier discretizations

Curves with rational chord-length parametrization

Approximation of Circular Arcs by C<sup>2</sup> Cubic Polynomial B-splines

Contact Info

Product

Resources

About