Affine generalised barycentric coordinates

• Shayne Waldron ^[1]
  1. [1] University of Auckland

     University of Auckland

     Nueva Zelanda

     
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 3, Nº. 2, 2011, págs. 209-226
• Idioma: inglés
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• Resumen

  • For a given set of points in Rᵈ, there may be many ways to write a point x in their affine hull as an affine combination of them. We show that there is a unique way which minimises the sum of the squares of the coefficients. It turns out that these coefficients, which are given by a simple formula, are affine functions of x, and so generalise the barycentric coordinates. These affine generalised barycentric coordinates have many nice properties, e.g., they depend continuously on the points, and transform naturally under symmetries and affine transformations of the points. Because of this, they are well suited to representing polynomials on polytopes. We give a brief discussion of the corresponding Bernstein-Bézier form and potential applications, such as finite elements and orthogonal polynomials.