C2 interpolation of spatial data subject to arc-length constraints using Pythagorean-hodograph quintic splines

Mathieu Huard (1) , Rida Farouki (2) , Nathalie Sprinsky (3) , Luc Biard (1)

1 MGMI - Modélisation Géométrique & Multirésolution pour l'Image
2 MAE - Department of Mechanical and Aeronautical Engineering
3 CEA-LETI - Commissariat à l'énergie atomique et aux énergies alternatives - Laboratoire d'Electronique et de Technologie de l'Information

In order to reconstruct spatial curves from discrete electronic sensor data, two alternative C2 Pythagorean-hodograph (PH) quintic spline formulations are proposed, interpolating given spatial data subject to prescribed constraints on the arc length of each spline segment. The first approach is concerned with the interpolation of a sequence of points, while the second addresses the interpolation of derivatives only (without spatial localization). The special structure of PH curves allows the arc-length conditions to be expressed as algebraic constraints on the curve coefficients. The C2 PH quintic splines are thus defined through minimization of a quadratic function subject to quadratic constraints, and a close starting approximation to the desired solution is identified in order to facilitate efficient construction by iterative methods. The C2 PH spline constructions are illustrated by several computed examples.

Mots clés: Space curves; Shape reconstruction; Arc-length constraints; Pythagorean-hodograph splines; Spatial data interpolation

Domaines: Géométrie différentielle [math.DG]

Mathieu Huard, Rida Farouki, Nathalie Sprinsky, Luc Biard. C2 interpolation of spatial data subject to arc-length constraints using Pythagorean-hodograph quintic splines. Graphical Models, 2014, 76 (1), pp.30-42. ⟨10.1016/j.gmod.2013.10.005⟩. ⟨hal-00983444⟩