Convolution Surfaces with Varying Radius: Formulae for Skeletons Made of Arcs of Circles and Line Segments

Chapter
Part of the Association for Women in Mathematics Series book series (AWMS, volume 12)

Abstract

In skeleton-based geometric modeling, convolution is an established technique: smooth surfaces around a skeleton made of curves are given as the level set of a convolution field. Varying the radius or making the surface scale sensitive along the skeleton are desirable features. This article provides the related necessary closed-form formulae of the convolution fields when the skeleton is made of arcs of circle and line segments. For the family of power inverse kernels, closed-form formulae are exhibited in terms of recurrence relationships. These are obtained by creative telescoping. This novel technique is described from a practitioner point of view so as to be applied to other families of kernels or skeleton primitives. The newly obtained formulae are applied to obtain convolution surfaces around G1 skeleton curves, some of them closed curves. Having arcs of circles in addition to line segments allows to demonstrably improve the visual quality of the generated surface with a lower number of skeleton primitives.

Notes

Acknowledgements

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 675789. We want to thank Shaoshi Chen, Bruno Salvy, Alin Bostan, Louis Dumont, Frédéric Chyzak, and Christoph Koutschan for several enlightening discussions on creative telescoping over the years, as well as making their fantastic software available.

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Copyright information

© The Author(s) and the Association for Women in Mathematics 2018

Authors and Affiliations

  • Alvaro Javier Fuentes Suárez
    • 1
  • Evelyne Hubert
    • 1
  1. 1.INRIA MéditerranéeSophia AntipolisFrance

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