While animation using barycentric coordinates or other automatic weight assignment methods has become a popular method for shape deformation, the global nature of the weights limits their use for real-time applications. We present a method that reduces the number of control points influencing a vertex to a user-specified number such that the deformations created by the reduced weight set resemb...

The cage-based deformation of a 3D object through generalized barycentric coordinates is a simple, e cient, e ective and hence widely used shape manipulation scheme. Editing vertices of the polyhedral cage induces a smooth space deformation of its interior; the vertices thus become control handles of the nal deformation. However, in some application elds, as medicine, constrained volumepreservi...

Generalizations of barycentric coordinates in two and higher dimensions have been shown to have a number of applications in recent years, including finite element analysis, the definition of Spatches (n-sided generalizations of Bézier surfaces), free-form deformations, mesh parametrization, and interpolation. In this paper we present a new form of d dimensional generalized barycentric coordinat...

The parametrization of 3-d meshes can be used in many fields of computer graphics. Mesh-texturing, mesh-retriangulation or 3-d morphing are only few applications for which a mesh parametrization is needed. Because, many polygonal surfaces are manifolds of genus 0 (topological equivalent to a sphere), we can apply a mapping, in which 2-d polar coordinates of a sphere can be directly transformed ...

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distance...

The present survey collects some recent results on barycentric interpolation showing the similarity to the corresponding Lagrange (or polynomial) theorems. Namely we state that the order of the Lebesgue constant for barycentric interpolation is at least log n; we state a Grünwald–Marcinkiewicz type theorem for the barycentric case; moreover we define a Bernstein type process for the barycentric...

A collection of recent papers reveals that linear barycentric rational interpolation with the weights suggested by Floater and Hormann is a good choice for approximating smooth functions, especially when the interpolation nodes are equidistant. In the latter setting, the Lebesgue constant of this rational interpolation process is known to grow only logarithmically with the number of nodes. But ...

The problem of hyperbolic incoming orbits for single-apparition comets is investigated. In this context the effect of non-gravitational acceleration on cometary dynamics was analyzed for the sample of 33 “hyperbolic” comets. The orbital elements of each cometary orbit were determined by the least squares procedure based on positional observations. These osculating orbital elements serve as a ba...

In this paper we present an easy computation of a generalized form of barycentric coordinates for irregular, convex n-sided polygons. Triangular barycentric coordinates have had many classical applications in computer graphics, from texture mapping to ray-tracing. Our new equations preserve many of the familiar properties of the triangular barycentric coordinates with an equally simple calculat...

Let K be a simplicial complex with vertex set V = {v1, . . . , vn}. The complex K is d-representable if there is a collection {C1, . . . , Cn} of convex sets in R d such that any subcollection {Ci1 , . . . , Cij} has a nonempty intersection if and only if {vi1 , . . . , vij} is a face of K. In 1967 Wegner proved that every simplicial complex of dimension d is (2d + 1)representable. He also conj...