### Abstract

A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.

Original language | English (US) |
---|---|

Pages (from-to) | 80-95 |

Number of pages | 16 |

Journal | Computer Aided Geometric Design |

Volume | 25 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2008 |

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### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Geometry and Topology
- Modeling and Simulation

### Cite this

*Computer Aided Geometric Design*,

*25*(2), 80-95. https://doi.org/10.1016/j.cagd.2007.05.005

**On the approximation order of tangent estimators.** / Albrecht, G.; Bécar, J. P.; Farin, G.; Hansford, D.

Research output: Contribution to journal › Article

*Computer Aided Geometric Design*, vol. 25, no. 2, pp. 80-95. https://doi.org/10.1016/j.cagd.2007.05.005

}

TY - JOUR

T1 - On the approximation order of tangent estimators

AU - Albrecht, G.

AU - Bécar, J. P.

AU - Farin, G.

AU - Hansford, D.

PY - 2008/2

Y1 - 2008/2

N2 - A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.

AB - A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.

UR - http://www.scopus.com/inward/record.url?scp=37249081652&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37249081652&partnerID=8YFLogxK

U2 - 10.1016/j.cagd.2007.05.005

DO - 10.1016/j.cagd.2007.05.005

M3 - Article

AN - SCOPUS:37249081652

VL - 25

SP - 80

EP - 95

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

SN - 0167-8396

IS - 2

ER -