On the Geometry and the Deformation of Shapes Represented by Piecewise Continuous Bézier Curves with Application to Shape Optimization

In this work, we develop a framework based on piecewise Bézier curves to plane shapes deformation and we apply it to shape optimization problems. We describe a general setting and some general result to reduce the study of a shape optimization problem to a finite dimensional problem of integration of a special type of vector field. We show a practical problem where this approach leads to efficient algorithms. In all the text below, E = ❘. In this text, we will define a set of manifolds, each point of such a manifold is a parametrized curves in E. 1 Bézier curves Bézier curves are usual objects in Computer Aided Geometric Design (CAGD) and have natural and straightforward generalization for surfaces and higher dimension geometrical objects. We focus here on plane curves even if most of results of this paper have a natural generalization in higher dimension. We will show that a Bézier curve is fully encoded by a set of points. This set of points forms the control polygon. This will lead us to a parametrization of curves by their control polygon giving to the set of Bézier (and piecewise Bézier) curves a manifold structure by diffeomorphism. This diffeomorphism is in fact linear and allows to studied space of curve as a vector space. Using this will be able to interpret deformations as infinitesimal curves of the same kind. The aim of this section is to fix notation and make the paper as self contained as possible. 1.1 Basic definitions Given P0, P1, . . . , PD ∈ E, we define: B (( P0, . . . , PD )