Efficient object location determination and error analysis based on barycentric coordinates

In this paper,we propose an efficient computational method for converting local coordinates to world coordinates using specially structured coordinate data.The problem in question is the computation of world coordinates of an object throughout a motion,assuming that we only know the changing coordinates of some fixed surrounding reference points in the local coordinate system of the object.The proposed method is based on barycentric coordinates;by taking the aforementioned static positions as the vertices of a polyhedron,we can specify the coordinates of the object in each step with the help of barycentric coordinates.This approach can significantly help us to achieve more accurate results than by using other possible methods.In the paper,we describe the problem and barycentric coordinate-based solution in detail.We then compare the barycentric method with a technique based on transformation matrices,which we also tested for solving our problem.We also present various diagrams that demonstrate the efficiency of our proposed approach in terms of precision and performance.

Distributed planar formation maneuvering of leader-follower networked systems via a barycentric coordinate-based approach

This paper presents a distributed planar leader-follower formation maneuver control strategy for multi-agent systems with different agent dynamic models.This method is based on the barycentric coordinate-based(BCB)control,which can be performed in the local coordinate frame of each agent with required local measurements.By exploring the properties of BCB Laplacians,a time-varying target formation can be BCB localizable by a sufficient number of leaders uniquely,and this formation is converted from a given nominal formation with geometrical similarity transformation.The proposed control laws can continuously maneuver collective single-and double-integrator agents to achieve a translation,scale,rotation,or even their compositions in various directions.For the formation shape control problem of multi-car systems with/without saturation constraints,the obtained control performance can preserve good robustness.Global stability is also proven by mathematical derivations and verified by numerical simulations.

Real-Time Tool for Affine Transformations of Two Dimensional IFS Fractal

This work introduces a novel tool for interactive, real-time affine transformations of two dimensional IFS fractals. The tool uses some of the nice properties of the barycentric coordinates that are assigned to the points that constitute the image ofa fractal, and thus enables any affine transformation of the affine basis, done by click-and-drag, to be immediately followed by the same affine transformation of the fractal. The barycentric coordinates can be relative to an arbitrary affine basis of ~2, but in order to have a better control over the fractal, a kind of minimal simplex that contains the fractal attractor is used.

ON TRIANGULAR C^1 SCHEMES: A NOVEL CONSTRUCTION

In this paper we present a C-1 interpolation scheme on a triangle. The interpolant assumes given values and one order derivatives at the vertices of the triangle. It is made up of partial interpolants blended with corresponding weight functions. Any partial interpolant is a piecewise cubics defined on a split of the triangle, while the weight function is just the respective barycentric coordinate. Hence the interpolant can be regarded as a piecewise quartic. We device a simple algorithm for the evaluation of the interpolant. It's easy to represent the interpolant with B-net method. We also depict the Franke's function and its interpolant, the illustration of which shows good visual effect of the scheme.

A CONFORMING QUADRATIC POLYGONAL ELEMENT AND ITS APPLICATION TO STOKES EQUATIONS

In this paper,we construct an H1-conforming quadratic finite element on convex polygonal meshes using the generalized barycentric coordinates.The element has optimal approximation rates.Using this quadratic element,two stable discretizations for the Stokes equations are developed,which can be viewed as the extensions of the P2-P0 and the Q2-(discontinuous)P1 elements,respectively,to polygonal meshes.Numerical results are presented,which support our theoretical claims.