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The spline space C r k (∆) attached to a subdivided domain ∆ of R d is the vector space of functions of class C r which are polynomials of degree ≤ k on each piece of this subdivision. Classical splines on planar rectangular grids play an important role in Computer Aided Geometric Design, and spline spaces over arbitrary subdivisions of planar domains are… (More)

Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a… (More)

Bounds on the dimension of trivariate spline spaces: A homological approach Abstract. We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by applying homological techniques. We… (More)

Characterization of bivariate hierarchical quartic box splines on a three-directional grid Abstract. We consider the adaptive refinement of bivariate quar-tic C 2-smooth box-spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be… (More)

A real irrational toric variety X is an analytic subset of the simplex associated to a finite configuration of real vectors. The positive torus acts on X by translation, and we consider limits of sequences of these translations. Our main result identifies all possible Hausdorff limits of translations of X as toric degenerations using elementary methods and… (More)

A real irrational toric variety X is an analytic subset of the simplex associated to a finite configuration of real vectors. The positive torus acts on X by translation, and we consider limits of sequences of these translations. Our main result identifies all possible Hausdorff limits of translations of X as toric degenerations using elementary methods and… (More)

Geometrically continuous splines for surfaces of arbitrary topology Powered by TCPDF (www.tcpdf.org) Abstract We analyze the space of geometrically continuous piecewise polynomial functions or splines for quadrangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions, we… (More)

We consider, from an algebro-geometric perspective, the problem of determining the dimension of the space of bivariate and trivariate piecewise polynomial functions (or splines) defined on triangular and tetrahedral partitions. Classical splines on planar rectangular grids play an important role in Computer Aided Geometric Design, and splines spaces over… (More)