Shape preserving approximation by spatial cubic splines

Pang, Kong Voon; Ong, B. H.

doi:10.1016/j.cagd.2009.07.001

Cited by 16 publications

(5 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are a lot of methods to calculate the discrete curvature and torsion [11][12][13][14][15]. In the paper, we choose the following approach to estimate the torsion at each point [14,15]…”

Section: Discrete Curvature and Torsion-based Methodsmentioning

confidence: 99%

Discrete curvature and torsion-based parameterization scheme for data points

Fang¹,

Wu²,

Liu³

2017

Proceedings of the 7th International Conference on Computer Engineering and Networks — PoS(CENet2017)

View full text Add to dashboard Cite

Parameterization for data points is a fundamental problem in the field of Computer Aided Geometric Design. Recently, an improved centripetal parameterization technique is proposed by Fang et al and its superiority to other common methods, such as uniform method, chord length method, centripetal method, Foley method and universal method, etc., is validated by a lot of numerical examples. In this article, a refined formula is firstly given to resolve the problem that the parameter values obtained by Fang's method are usually a little bigger than the optimum ones. Furthermore, as enlightened by the local differential geometric properties of parametric curves, a novel parameterization scheme is put forward by introducing the discrete curvature and torsion information at each point. As indicated by numerical experiments, the deviation measured by a curvature and Euler distance-based criterion between the interpolation B-spline curve obtained by our method and the polyline constructed by the points are smaller than the ones between the curves obtained by Fang's method, the chord length method, the standard centripetal method and the polyline. The proposed algorithm is applicable to both 2D and 3D points and has a distinct advantage for 3D points as compared with the three aforementioned methods.

show abstract

Section: Discrete Curvature and Torsion-based Methodsmentioning

confidence: 99%

Discrete curvature and torsion-based parameterization scheme for data points

Fang¹,

Wu²,

Liu³

2017

Proceedings of the 7th International Conference on Computer Engineering and Networks — PoS(CENet2017)

View full text Add to dashboard Cite

show abstract

“…More details can be found in [28]. 6 It is known that the curve is closer to the control point when its associated weight is greater. We now consider the point which has the maximum distance to the planes P 1 = p 0 p 2 p 3 and P 2 = p 0 p 1 p 3 respectively.…”

Section: ) P(t) Has No Singular Points Andmentioning

confidence: 99%

“…One basic problem in the study of parametric curves is to approximate the curve with lower degree curve segments. For a given digital curve, there exist methods to find such approximate curves efficiently [3,4,5,6]. If the curve is given by explicit expressions, either parametric or implicit, these methods are still usable.…”

Section: Introductionmentioning

confidence: 99%

Certified approximation of parametric space curves with cubic B-spline curves

Shen

Yuan

Gao

2012

Computer Aided Geometric Design

View full text Add to dashboard Cite

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given precision and preserve the geometric features of the given curve such as the topology, singular points, etc. The approximated curve is divided into segments called quasi-cubic Bézier curve segments which have properties similar to a cubic rational Bézier curve. And the approximate curve is naturally constructed as the associated cubic rational Bézier curve of the control tetrahedron of a quasi-cubic curve. A novel optimization method is proposed to select proper weights in the cubic rational Bézier curve to approximate the given curve. The error of the approximation is controlled by the size of its tetrahedron, which converges to zero by subdividing the curve segments. As an application, approximate implicit equations of the approximated curves can be computed. Experiments show that the method can approximate space curves of high degrees with high precision and very few cubic Bézier curve segments.

show abstract

“…Although somewhat more complicated to implement, B-splines may be preferred to cubic splines, due to its robustness to bad data, and ability to preserve monotonicity and convexity. A recent paper [99] describes a computationally efficient approach for cubic B-splines.…”

Section: Remarks On Spline Interpolationmentioning

confidence: 99%

Implied Volatility Surface: Construction Methodologies and Characteristics

Homescu¹

2011

SSRN Journal

View full text Add to dashboard Cite

The implied volatility surface (IVS) is a fundamental building block in computational finance. We provide a survey of methodologies for constructing such surfaces. We also discuss various topics which influence the successful construction of IVS in practice: arbitrage-free conditions in both strike and time, how to perform extrapolation outside the core region, choice of calibrating functional and selection of numerical optimization algorithms, volatility surface dynamics and asymptotics. *

show abstract

Shape preserving approximation by spatial cubic splines

Cited by 16 publications

References 11 publications

Discrete curvature and torsion-based parameterization scheme for data points

Discrete curvature and torsion-based parameterization scheme for data points

Certified approximation of parametric space curves with cubic B-spline curves

Implied Volatility Surface: Construction Methodologies and Characteristics

Contact Info

Product

Resources

About