Barycentric coordinate system

In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex. Every point has barycentric coordinates, and their sum is not zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity. Barycentric coordinates were intr

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This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L-2-norm and the H-1-seminorm. 2022 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2022

Local Barycentric Coordinates

Sofien Bouaziz, Bailin Deng

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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