It is well known that in two or more variables Bernstein polynomi-als do not preserve convexity. Here we introduce two variations, one stronger than the classical notion, the other one weaker, which are preserved. Moreover, a weaker suucient condition for the monotony of subsequent Bernstein polynomials is given. linearly independent, in the course of which d has to be greater than or equal to ...

In this paper, we present the first phase of our effort to build a bone density atlas. We adopted a tetrahedral mesh structure to represent anatomical structures. We propose an efficient and automatic algorithm to construct the tetrahedral mesh from contours in CT images corresponding to the outer bone surfaces and boundaries between compact bone, spongy bone, and medullary cavity. We approxima...

Two planar triangulations with a correspondence between two vertex sets are compatible (isomorphic) if they are topologically equivalent. This work presents a simple and robust method for morphing two compatible planar triangulations with identical convex boundaries that locally preserves the intrinsic geometric properties of triangles throughout the morph. The method is based on the barycentri...

Let Xn := {xi}ni=0 be a given set of (n + 1) pairwise distinct points in R (called nodes or sample points), let P = conv(Xn), let f be a convex function with Lipschitz continuous gradient on P and λ := {λi}ni=0 be a set of barycentric coordinates with respect to the point set Xn. We analyze the error estimate between f and its barycentric approximation:

It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no real poles and arbitrarily high approximation orders on any real interval, regardless of th...

Let ∆ be the area of the fundamental triangle ABC of barycentric coordinates and let α = cotA, β = cotB, γ = cotC. The vectors vi = [xi, yi, zi] (i = 1, 2) have the scalar product 2∆(αx1x2 + βy1y2 + γz1z2). This fact implies all important formulas about metrical relations of points and lines. The main and probably new results are Theorems 1 and 8.

j j j k z T z T z = = ∑ where T is a 2D similarity transformation. Proof: Similarity transformations can be represented using a linear polynomial over the complex plane in the following way. If the similarity transformation T consists of rotation by positive angle θ, uniform scale s and a translation t = tx + ity, then according to the rules of complex numbers: ( ) ( ) i T x iy T z se z t z θ +...

Associated to any finite flag complex L there is a right-angled Coxeter group WL and a contractible cubical complex ΣL on which WL acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a triangulation of Sn−1, then ΣL is a contractible n-manifold. We establish vanishing (in a certain range) of the reduced `2-homology of ΣL in the case where L is the ...

In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of...