In this work we focus on approximations of continuous harmonic functions by discrete harmonic functions based on the discrete Laplacian in a triangulation of a point set. We show how the choice of edge weights based on generalized barycentric coordinates influences the approximation quality of discrete harmonic functions. Furthermore, we consider a varying point set to demonstrate that generali...

Given three regular space curves r1(t), r2(t) r3(t), t ∈ [ 0, 1 ] that define a curvilinear triangle, we consider the problem of constructing a C triangular surface patch R(u1, u2, u3) bounded by these three curves, such that they are geodesics of the constructed surface. Results from a prior study [6] concerned with tensor–product patches are adapted to identify constraints on the given curves...

The paper presents a visualization technique that facilitates and eases analyses of interestingness measures with respect to their properties. Detection of properties possessed by these measures is especially important when choosing a measure for KDD tasks. Our visual-based approach is a useful alternative to often laborious and time consuming theoretical studies, as it allows to promptly perce...

This paper presents an algorithm for material interface reconstruction for data sets where fractional material information is given as a percentage for each element of the underlying grid. The reconstruction problem is transformed to a problem that analyzes a dual grid, where each vertex in the dual grid has an associated barycentric coordinate tuple that represents the fraction of each materia...

Let SN(P) be the poset obtained by adding a dummy vertex on each diagonal edge of the N’s of a finite poset P. We show that SN(SN(P)) is N-free. It follows that this poset is the smallest N-free barycentric subdivision of the diagram of P, poset whose existence was proved by P.A. Grillet. This is also the poset obtained by the algorithm starting with P0 := P and consisting at step m of adding a...

We present a new perspective on the Floater–Hormann interpolant. This interpolant is rational of degree (n, d), reproduces polynomials of degree d, and has no real poles. By casting the evaluation of this interpolant as a pyramid algorithm, we first demonstrate a close relation to Neville’s algorithm. We then derive an O(nd) algorithm for computing the barycentric weights of the Floater–Hormann...

2 Mathematical Preliminaries 3 2.1 Barycentric Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Inscribed Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Bezier Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Derivatives . . . . . . . . . . ....

Let ABC be a fundamental triangle with the area . For a circle K the powers of vertices A, B, C with regard to K divided by 2 are said to be the barycentric coordinates of K with respect to triangle ABC. This paper gives some theory and applications of these coordinates.

Abstract: Trivariate barycentric coordinates can be used both to express a point inside a tetrahedron as a convex combination of the four vertices and to linearly interpolate data given at the vertices. In this paper we generalize these coordinates to convex polyhedra and the kernels of star-shaped polyhedra. These coordinates generalize in a natural way a recently constructed set of coordinate...

The tree complex is a simplicial defined in recent work of Belk, Lanier, Margalit, and Winarski with applications to mapping class groups dynamics. This article introduces connection between this setting the convex polytopes known as associahedra cyclohedra. Specifically, we describe characterization these using planar embeddings trees show that barycentric subdivision polyhedral cell for which...