On the injectivity of Wachspress and mean value mappings between convex polygons

Wachspress and mean value coordinates are two generalizations of triangular barycentric coordinates to convex polygons and have recently been used to construct mappings between polygons, with application to curve deformation and image warping. We show that Wachspress mappings between convex polygons are always injective but that mean value mappings can fail to be so in extreme cases.     

Interpolation error estimates for mean value coordinates over convex polygons

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach π.     

Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

On Wachspress pentagonal patches

Wachspress quadrilateral patches have been recently studied from the point of view of applications to surface modelling in CAGD [1], [3], [4]. Some more applications for defining barycentric coordinates for arbitrary polygons have also been presented in [5] [9]. The purpose of the present paper is to introduce non-negative Wachspress rational basis functions for surface modelling on pentagonal partitions. Interpolation formula for function values and directional derivatives at the vertices of pentagon has been presented. Conditions for C¹– continuity of the composite surface have also been studied in the paper. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

凸多边形上的Bezier曲面片

本文建立了正多边形上的重心坐标，并构造了多边形上的Ｂｅｚｉｅｒ曲面片、     

Barycentric coordinates for convex polytopes

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, theadjoint of the polytope, generalizes a previous two-dimensional construction described by Wachspress. The barycentric coordinate functions for the polytope are rational combinations of adjoints of various dual cones associated with the polytope.     

Error estimates for generalized barycentric interpolation

We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.     

On the Structure of Multiple Translational Tilings by Polygonal Regions

We consider polygons with the following ``pairing property': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon K tiles multiply the plane when translated at the locations Λ , where Λ is a multiset in the plane. The pairing property of K makes this question particularly amenable to Fourier analysis. As a first application of our approach we establish a necessary and sufficient condition for K to tile with a given lattice Λ . (This was first found by Bolle for the case of convex polygons—notice that all convex polygons that tile, necessarily have the pairing property and, therefore, our theorems apply to them.) Our main result is a proof that a large class of such polygons tile multiply only quasi-periodically, which for us means that Λ must be a finite union of translated two-dimensional lattices in the plane. For the particular case of convex polygons we show that all convex polygons which are not parallelograms tile multiply only quasi-periodically, if at all. Received February 24, 1999, and in revised form August 26, 1999, and October 9, 1999.     

Minimization of convex functions on the convex hull of a point set

A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in R ⁿ is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal to m. The method is based on a new procedure for finding the projection of the gradient of the objective function onto a simplicial cone in R ^m , which is the tangent cone at the current point to the simplex defined by the usual constraints on barycentric coordinates. It is shown that this projection can be computed in O(m log m) operations. For strictly convex quadratic functions, the basic method can be refined to a noniterative method terminating with the optimal solution.     

Generalized polar coordinates on Lie groups and numerical integrators

Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps. In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres.     

The mean of circumscribed polygons

The mean of circumscribed polygons of a convex body in \Bbb R ²\Bbb R ^2 is again a convex body. The corresponding mapping of convex bodies is an endomorphism, and we characterize the injective case. It is shown that this endomorphism is almost always injective.     

Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝ^{n + 1} remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1 , and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc.     

A characterization of affine length and asymptotic approximation of convex discs

It is shown that every equi-affine invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, area, and affine length. Asymptotic formulae for approximation of convex discs by polygons are derived, extending results of L. Fejes Tóth from smooth convex discs to general convex discs.     

The mean value of the area of polygons circumscribed about a plane convex body

Inner parallel bodies are used to prove that the mean area of polygons circumscribed about a convex bodyK of given area is minimum whenK is a circle.     

Constructing convex solutions via Perron’s method

Abstract In this paper we construct convex solutions for certain elliptic boundary value problems via Perron’s method. The solutions constructed are weak solutions in the viscosity sense, and our construction follows work of Ishii (Duke Math. J., 55 (2) 369–384, 1987). The same general approach appears in work of Andrews and Feldman (J. Differential Equations, 182 (2) 298–343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves. The time independent special case of their work leads to a one dimensional elliptic result which we extend to two dimensions. Similar results are required to extend their theory of nonlocal geometric flows to surfaces. The two dimensional case is essentially different from the one dimensional case and involves a regularity result (cf. Theorem 3.1), which has independent interest. Roughly speaking, given an arbitrary convex function (which is not smooth) supported at one point by a smooth function of prescribed Hessian (which is not convex), one must construct a third function that is both convex and smooth and appropriately approximates both of the given functions. Keywords: Viscosity solutions, Elliptic partial differential equations, Perron procedure, Convexity, Regularity, Fully nonlinear, Monge-Ampere Mathematics Subject Classification (2000:) 35J60, 53A05, 52A15, 26B05     

Decomposing a simple polygon into pseudo-triangles and convex polygons

In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudo-triangles and convex polygons. We call the resulting decomposition PT-convex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolygons of one of two types has the potential to reduce the complexity of the resulting decomposition considerably.The problem of decomposing a simple polygon into the least number of convex polygons has been considered. We extend a dynamic-programming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudo-triangles are allowed. Our algorithm determines such a decomposition in O(n³) time and space, where n is the number of the vertices of the polygon.     

Pebble Sets in Convex Polygons

Lukács and András posed the problem of showing the existence of a set of n−2 points in the interior of a convex n-gon so that the interior of every triangle determined by three vertices of the polygon contains a unique point of S. Such sets have been called pebble sets by De Loera, Peterson, and Su. We seek to characterize all such sets for any given convex polygon in the plane. We first consider a certain class of pebble sets, called peripheral because they consist of points that lie close to the boundary of the polygon. We characterize all peripheral pebble sets, and show that for regular polygons, these are the only ones. Though we demonstrate examples of polygons where there are other pebble sets, we nevertheless provide a characterization of the kinds of points that can be involved in non-peripheral pebble sets. We furthermore describe algorithms to find such points.     

A decomposition theorem for convexity spaces

We utilize the unifying framework of families of convexity spaces for the treatment of various notions of planar convexity and the associated convex hulls. Our major goal is to prove the refinement and decomposition theorems for families of convexity spaces. These general theorems are then applied to two examples: restricted-oriented convex sets andNESW-convex sets. The applications demonstrate the usefulness of these general theorems, since they give rise to simple algorithms for the computation of the associated convex hulls of polygons.     

Polygons with Parallel Opposite Sides

In this paper we consider convex planar polygons with parallel opposite sides. These polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For such polygons, we define discrete versions of the area evolute, central symmetry set, equidistants, and area parallels and show that they behave quite similarly to their smooth counterparts.