Seminars

In CAD systems, a piecewise polynomial function is behind any curve, surface or scalar field representation. Thus, it is important to analyse the properties of the spaces of piecewise polynomial functions.In this thesis, we study commutative algebra tools that can be used to analyze the dimension of piecewise polynomial spaces, and to construct bases for them. We test the methods that we produce to model free form surfaces and for numerical analysis computations. The main motivation for the concept of geometric continuity is the construction of multi-patches surfaces and scalar fields. The main challenge in this kind of surfaces is to handle areas of the surface around vertices with a number of neighboring patches different from 4 (that we call Extraordinary vertices). In these regions, the usual gluing methods will cause the appearance of singularities. Geometric continuity is a special way to glue two 3d surface patches along their common edge in a multi-patch surface, and that produces smooth surfaces even around extraordinary vertices. The geometric continuity gluing condition is expressed in terms of linear relations between the parametrizations of the surfaces along there junction edges. The coefficients of those relations are called the gluing data, and there choice is crucial for the smoothness of the resulting surface. The gluing data that we propose are spline functions that respect smoothness constraint such as the vertex enclosure constraint. We explain our choice by providing a formula that the gluing data have to respect at each extraordinary vertex.We require that the Geometrically continuous spline (We call Gsplines the Geometrically continuous splines) spaces that we produce to be able to interpolate any given positions of the vertices of its corresponding mesh. This is what we call the separability condition. We describe conditions on the gluing data that allows the space to be separable, and give a list of examples of such a gluing data. The manuscript also describe a “piecing scheme” that allows to produce basis for the space of Gsplines.We have addressed the possibility of extending the existing homology methods to analyse the dimension of spline space with geometric continuity conditions. These extensions provide many formulas that expresses the dimensions of our spline spaces by means of other homology groups.

Our analyse of this space leads to three applications: The first one is an algorithm that given a mesh, produces a smooth surface that approximates it. This algorithm is based on the projection of the Approximate Catmull-Clark surface on the space of splines that we produce. The two other tests are on smooth surfaces reconstruction and IsoGeometric analysis.