Let $$T=\{\triangle _1,\ldots ,\triangle _n\}$$ be a set of n triangles in $$\mathbb {R}^3$$ with pairwise-disjoint interiors, and let B convex polytope constant number faces. For each i, $$C_i = \triangle _i \oplus r_i B$$ denote the Minkowski sum $$\triangle _i$$ copy scaled by $$r_i>0$$ . We show that if scaling factors $$r_1, \ldots , r_n$$ are chosen randomly then expected complexity union...

The F-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the F-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the " F = M Conjecture " of Armstrong which predicts a surprising relation between the F-triangle and the Möbius function of his m-divisible partition poset associated t...

In this paper a technique for the construction of quartic polynomial B-spline patches fitting on analytical surfaces and triangle meshes is presented. The input data are curvature values and principal directions at a given surface point which can be computed directly, if the surface is represented by a vector function. In the case of discrete surface representation, i.e. on a triangle mesh the ...

This paper discusses a formation control problem in which a target formation is defined with both distance and signed area constraints. The control objective is to drive spatially distributed agents to reach a unique target rigid formation shape (up to rotation and translation) with desired inter-agent distances. We define a new potential function by incorporating both distance terms and signed...

It is observed that the Powell-Sabin 12-split triangle is re nable since the same split of the 4 similar subtriangles of a triangle contains the lines of split of the original triangle. This property of the split is the key to the existence of a subdivision scheme, for the evaluation of the C quadratic spline on the split which interpolates function and gradient values at the 3 vertices of the ...

It is observed that the Powell-Sabin 12-split triangle is reen-able since the same split of the 4 similar subtriangles of a triangle contains the lines of split of the original triangle. This property of the split is the key to the existence of a subdivision scheme, for the evaluation of the C 1 quadratic spline on the split which interpolates function and gradient values at the 3 vertices of t...

Hecke proved the meromorphic continuation of a Dirichlet series associated to the lattice points in a triangle with a real quadratic slope and found the possible poles in terms of the fundamental unit. An analogous result is proven for certain elliptical cones where now the poles are determined by the spectrum of the Laplacian on an arithmetic Riemann surface. 1 Some results of Hardy-Littlewood...

In this paper, the intersection of bivariate orthogonal polynomials on triangle patches will be investigated. The result is interesting by its own but also has important applications in the theory of a posteriori error estimation for finite element discretizations with p-refinement, i.e., if the local polynomial degree of the test and trial functions is increased to improve the accuracy. A tria...

Abstract In this paper, we address the problem of constructing $$C^2$$ C 2 cubic spline functions on a given arbitrary triangulation $${\mathcal {T}}$$ T . To end, endow every triangle with Wang–Shi macro-structure. The space such refined has stable dim...