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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 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework… (More)

- Bernard Mourrain, Nelly Villamizar
- J. Symb. Comput.
- 2013

The spline space C k (∆) attached to a subdivided domain ∆ of R is the vector space of functions of class C 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 now… (More)

- Matteo Gallet, Christoph Koutschan, Zijia Li, Georg Regensburger, Josef Schicho, Nelly Villamizar
- Math. Comput.
- 2017

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)

- Bernard Mourrain, Raimundas Vidunas, Nelly Villamizar
- Computer Aided Geometric Design
- 2016

- Nelly Villamizar, Angelos Mantzaflaris, Bert Jüttler
- Computer Aided Geometric Design
- 2016

We consider the adaptive refinement of bivariate quartic C-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 called the space of special quartics. Given a finite sequence (G)l=0,...,N of dyadically refined grids, we… (More)

- Bernard Mourrain, Nelly Villamizar
- Mathematics in Computer Science
- 2014

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 give an insight of different ways of approaching this problem by exploring its… (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)

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)

- Ahmed Blidia, Bernard Mourrain, Nelly Villamizar
- Computer Aided Geometric Design
- 2017

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