We take a geometric point of view on the recent result by Brenti and Welker, who showed that the roots of the f -polynomials of successive barycentric subdivisions of a finite simplicial complex X converge to fixed values dependinig only on the dimension of X . We show that these numbers are roots of a certain polynomial whose coefficients can be computed explicitely. We observe and prove an in...

The class of Strongly Signable partially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs’ recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flag h-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable pose...

In this paper we construct and study a new 15-vertex triangulationX of the complex projective plane CP. The automorphism group of X is isomorphic to S4 ×S3. We prove that the triangulation X is the minimal by the number of vertices triangulation of CP admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for simplices of X and show that the automorphism...

The coefficients of the chain polynomial a finite poset enumerate chains in by their number elements. polynomials partition lattices and standard type \(B\) analogues are shown to have only real roots. real-rootedness is conjectured for all geometric be preserved pyramid prism operations on Cohen-Macaulay posets. As result, new families convex polytopes whose barycentric subdivisions real-roote...

Yesterday, we introduced barycentric coordinates and de Casteljau’s algorithm. Today, we want to go more in depth into the mechanics of de Casteljau’s algorithm, and understand some of the nuances of the algorithm. We also want to discuss the efficiency of this algorithm in creating the curve. The algorithm’s power is not necessarily in defining a polynomial curve, but in how the algorithm can ...

We discuss a natural way to deene barycentric coordinates associated with circular arcs. This leads to a theory of Bernstein-B ezier polynomials which parallels the familiar interval case, and which has close connections to trigonometric polynomials. x1. Introduction Bernstein-B ezier (BB-) polynomials deened on an interval are useful tools for constructing piecewise functional and parametric c...

Improving data visualization paradigms for performance management is an orphaned area of tool development. Tool vendors avoid investing in development if they see no demand, while capacity planners and performance analysts do not demand what they have not conceived. We attempt to cut this Gordian knot with ’Barry’; a 3D performance visualization suite based on barycentric coordinates. Potential...

Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two-dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some ...

Virtual reality and haptic systems use geometric transformations with points represented in homogeneous coordinates extensively. In many cases interpolation and barycentric coordinates are used. However, developers do not fully use properties of projective representation to make algorithms stable, robust and faster. This paper describes geometric transformations and principle of duality which e...