Coordonnées barycentriques

En géométrie affine, les coordonnées barycentriques d'un point par rapport à un repère barycentrique sont une famille de poids permettant de définir ce point comme un barycentre. Définitions Base affine Repère affine Une famille finie (P,…,P) de points d'un espace affine E est dite affinement libre, ou encore ces points sont dits affinement indépendants, quand aucun des points P{{ind|i}} n'appartient au sous-espace affine engendré par les k autres points. Dans le cas contraire il est dit affinement lié. Par exemple deux points distincts forment une famille affinement libre, de même 3 points non alignés, et 4 points non coplanaires. De façon équivalente une famille finie (P,…,P) est affinement libre quand l'une des conditions suivantes est réalisée :

aucun des points P{{ind|i}} n'est barycentre des autres points ;
la famille de vecteurs \left(\overrightarrow{P_0P_1},\overrightarrow{P_0P_2},\dots,\overrightarrow{P_0P_k}\right)

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Juan Cao, Xiaodong Wei

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Barycentric coordinates yield a powerful and yet simple paradigm to interpolate data values on polyhedral domains. They represent interior points of the domain as an affine combination of a set of control points, defining an interpolation scheme for any function defined on a set of control points. Numerous barycentric coordinate schemes have been proposed satisfying a large variety of properties. However, they typically define interpolation as a combination of all control points. Thus a local change in the value at a single control point will create a global change by propagation into the whole domain. In this context, we present a family of local barycentric coordinates (LBC), which select for each interior point a small set of control points and satisfy common requirements on barycentric coordinates, such as linearity, non-negativity, and smoothness. LBC are achieved through a convex optimization based on total variation, and provide a compact representation that reduces memory footprint and allows for fast deformations. Our experiments show that LBC provide more local and finer control on shape deformation than previous approaches, and lead to more intuitive deformation results.

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