Polynomial-based non-uniform interpolatory subdivision with features control

Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that the most convenient parameter values may be chosen as well as the intervals for insertion. Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control

Complexity of triangulations of the projective space

It is known that any two triangulations of a compact 3-manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3-dimensional projective space, in terms of the number of tetrahedra.Comment: 10 pages, 3 figures. Revised version, to appear in Top. App

How to make a triangulation of S^3 polytopal

We introduce a numerical isomorphism invariant p(T) for any triangulation T of S^3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable then p(T) is `small' in the sense that we obtain a linear upper bound for p(T) in the number n=n(T) of tetrahedra of T. Conversely, if p(T) is `small' then T is `almost' polytopal, since we show how to transform T into a polytopal triangulation by O((p(T))^2) local subdivisions. The minimal number of local subdivisions needed to transform T into a polytopal triangulation is at least $\frac{p(T)}{3n}-n-2$. Using our previous results [math.GT/0007032], we obtain a general upper bound for p(T) exponential in n^2. We prove here by explicit constructions that there is no general subexponential upper bound for p(T) in n. Thus, we obtain triangulations that are `very far' from being polytopal. Our results yield a recognition algorithm for S^3 that is conceptually simpler, though somewhat slower, as the famous Rubinstein-Thompson algorithm.Comment: 24 pages, 17 figures. Final versio

Dynamical partitions of space in any dimension

Topologically stable cellular partitions of D dimensional spaces are studied. A complete statistical description of the average structural properties of such partition is given in term of a sequence of D/2-1 (or (D-1)/2) variables for D even (or odd). These variables are the average coordination numbers of the 2k-dimensional polytopes (2k < D) which make the cellular structure. A procedure to built D dimensional space partitions trough cell-division and cell-coalescence transformations is presented. Classes of structures which are invariant under these transformations are found and the average properties of such structures are illustrated. Homogeneous partitions are constructed and compared with the known structures obtained by Voronoi partitions and sphere packings in high dimensions.Comment: LaTeX 5 eps figures, submetted to J. Phys.

On the dimension of spline spaces on planar T-meshes

We analyze the space of bivariate functions that are piecewise polynomial of bi-degree \textless{}= (m, m') and of smoothness r along the interior edges of a planar T-mesh. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are enough regular. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness