Disputas: Nelly Yazmin Villamizar Villamizar
M. Sc. Nelly Yazmin Villamizar Villamizar ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.: Algebraic Geometry for Splines
Nelly Yazmin Villamizar Villamizar
Tid og sted for prøveforelesning
Professor Rosa Maria Miró-Roig, University of Barcelona
Professor André Galligo, Université Nicé Sophia Antipolis
Professor Tom Lyche, Centre of Mathematics for Applications, University of Oslo
Leder av disputas
Instituttleder Arne Huseby, Matematisk institutt, Universitet i Oslo
- Professor Ragni Piene, Matematisk institutt, Universitet i Oslo
- Professor Kristian Ranestad, Matematisk institutt, Universitet i Oslo
Spline functions and parametric Bézier curves and surfaces are widely used to represent geometric objects in Computer Aided Geometric Design (CAGD); they are used in animation software such as Adobe Flash, as well as for design, testing and manufacture of airplane wings. Splines on triangular meshes are very useful for modeling surfaces of arbitrary shape, apart from their applications to numerical analysis and to the solution of partial differential equations.
In this thesis we study, from the algebraic geometry perspective, two problems in CAGD: the problem of constructing and analyzing piecewise polynomials, or spline functions, on triangulated regions in two and three dimensions, and how toric degenerations of real toric varieties are related to the regular control surfaces of toric Bézier patches.
The main contributions of this work are the new formulas for upper and lower bounds on the dimension of spline spaces, and the study of the ring structure of planar spline spaces. We explore the significance for modeling of the control points and edges that form the control net of Bézier surfaces; we put in a more general setting previous results, which also lead us a new and elementary interpretation of the secondary polytope associated to a point configuration.
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