Barycentric coordinates are a fundamental concept in computer graphics and geometric modeling. We extend the geometric construction of Floater’s mean value coordinates [8,11] to a general form that is capable of constructing a family of coordinates in a convex 2D polygon, 3D triangular polyhedron, or a higher-dimensional simplicial polytope. This family unifies previously known coordinates, inc...

In this paper, kinematic models for multiple manipulator space robotic systems are developed, as functions of body-fixed barycentric vectors. These models are used to define workspaces for single and multi-manipulator freefloating systems. It is shown that following the capture of a large payload, the location of these workspaces in space changes, and their size is reduced. These effects, commo...

An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described. The key to this extension is the construction, for a given convex polytope, of a unique polynomial associated with that polytope. This polynomial, the adjoint of the polytope, generalizes a previous two-dimensional construction described by Wachspress. The barycentric coordina...

The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization ...

Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two-dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real-valued barycentric coordin...

We present a method for naturally and continuously morphing two simple planar polygons with corresponding vertices in a manner that guarantees that the intermediate polygons are also simple. This contrasts with all existing polygon morphing schemes who cannot guarantee the non-self-intersection property on a global scale, due to the heuristics they employ. Our method achieves this property by r...

Based on permutation enumeration of the symmetric group and ‘generalized’ barycentric coordinates on arbitrary convex polytope, we develop a technique to obtain symmetrization procedures for functions that provide a unified framework to derive new Hermite-Hadamard type inequalities. We also present applications of our results to the Wright-convex functions with special emphasis on their key rol...

We identify the images of the comparision maps from ordinary homology and Sobolev homology, respectively, to the `1-homology of a wordhyperbolic group with coefficients in complete normed modules. The underlying idea is that there is a subdivision procedure for singular chains in negatively curved spaces that is much more efficient (in terms of the `1-norm) than barycentric subdivision. The res...

In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function f , which for an arbitrary vertex v of the barycentric subdivision B of simplex K assigns some vertex from a face of K which contains v, we can find a simplex S of B which satisfies f (S) = K (see [10]). The notati...

Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.