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Computing Isophotes on Free-Form Surfaces Based on Support Function Approximation

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Mathematics of Surfaces XIII (Mathematics of Surfaces 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5654))

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Abstract

The support function of a free-form-surface is closely related to the implicit equation of the dual surface, and the process of computing both the dual surface and the support function can be seen as dual implicitization. The support function can be used to parameterize a surface by its inverse Gauss map. This map makes it relatively simple to study isophotes (which are simply images of spherical circles) and offset surfaces (which are obtained by adding the offsetting distance to the support function).

We present several classes of surfaces which admit a particularly simple computation of the dual surfaces and of the support function. These include quadratic polynomial surfaces, ruled surfaces with direction vectors of low degree and polynomial translational surfaces of bidegree (3,2).

In addition, we use a quasi-interpolation scheme for bivariate quadratic splines over criss-cross triangulations in order to formulate a method for approximating the support function. The inverse Gauss maps of the bivariate quadratic spline surfaces are computed and used for approximate isophote computation. The approximation order of the isophote approximation is shown to be 2.

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Author information

Authors and Affiliations

  1. Johannes Kepler University, Linz, Austria

    M. Aigner & B. Jüttler

  2. University of Cantabria, Santander, Spain

    L. Gonzalez-Vega

  3. University of Siena, Italy

    M. L. Sampoli

Authors
  1. M. Aigner
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  2. L. Gonzalez-Vega
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  3. B. Jüttler
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  4. M. L. Sampoli
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Editor information

Editors and Affiliations

  1. Computer Vision and Pattern Recognition Group,Computer Science, University of York Heslington, YO10-5DD, York, United Kingdom

    Edwin R. Hancock

  2. School of Computer Science, Cardiff University, Cardiff, UK

    Ralph R. Martin

  3. Numerical Geometry Ltd, 26 Abbey Lane, Lode, CB5 9EP, Cambridge, UK

    Malcolm A. Sabin

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© 2009 Springer-Verlag Berlin Heidelberg

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Aigner, M., Gonzalez-Vega, L., Jüttler, B., Sampoli, M.L. (2009). Computing Isophotes on Free-Form Surfaces Based on Support Function Approximation. In: Hancock, E.R., Martin, R.R., Sabin, M.A. (eds) Mathematics of Surfaces XIII. Mathematics of Surfaces 2009. Lecture Notes in Computer Science, vol 5654. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03596-8_1

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  • DOI: https://doi.org/10.1007/978-3-642-03596-8_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03595-1

  • Online ISBN: 978-3-642-03596-8

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