Volume 7, Issue 3
Interpolation by $G^2$ Quintic Pythagorean-Hodograph Curves

Gašper Jaklič, Jernej Kozak, Marjeta Krajnc, Vito Vitrih & Emil Žagar

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 374-398.

Published online: 2014-07

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  • Abstract

In this paper, the $G^2$ interpolation by Pythagorean-hodograph (PH) quintic curves in $\mathbb{R}^d$, $d ≥2$, is considered. The obtained results turn out as a useful tool in practical applications. Independently of the dimension $d$, they supply a $G^2$ quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns. Although several solutions might exist, the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case. The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method. Numerical examples confirm the efficiency of the proposed method.

  • Keywords

Pythagorean-hodograph curve, Hermite interpolation, geometric continuity, nonlinear analysis, homotopy.

  • AMS Subject Headings

65D05, 65D17

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-374, author = {}, title = {Interpolation by $G^2$ Quintic Pythagorean-Hodograph Curves}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {3}, pages = {374--398}, abstract = {

In this paper, the $G^2$ interpolation by Pythagorean-hodograph (PH) quintic curves in $\mathbb{R}^d$, $d ≥2$, is considered. The obtained results turn out as a useful tool in practical applications. Independently of the dimension $d$, they supply a $G^2$ quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns. Although several solutions might exist, the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case. The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method. Numerical examples confirm the efficiency of the proposed method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1314nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5880.html} }
TY - JOUR T1 - Interpolation by $G^2$ Quintic Pythagorean-Hodograph Curves JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 374 EP - 398 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1314nm UR - https://global-sci.org/intro/article_detail/nmtma/5880.html KW - Pythagorean-hodograph curve, Hermite interpolation, geometric continuity, nonlinear analysis, homotopy. AB -

In this paper, the $G^2$ interpolation by Pythagorean-hodograph (PH) quintic curves in $\mathbb{R}^d$, $d ≥2$, is considered. The obtained results turn out as a useful tool in practical applications. Independently of the dimension $d$, they supply a $G^2$ quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns. Although several solutions might exist, the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case. The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method. Numerical examples confirm the efficiency of the proposed method.

Gašper Jaklič, Jernej Kozak, Marjeta Krajnc, Vito Vitrih & Emil Žagar. (2020). Interpolation by $G^2$ Quintic Pythagorean-Hodograph Curves. Numerical Mathematics: Theory, Methods and Applications. 7 (3). 374-398. doi:10.4208/nmtma.2014.1314nm
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