Curves and surfaces

Previous lectures have laid the foundations of algorithmic geometry. The following lectures will focus on the construction of computer models representing the complex geometric shapes that can be digitized today, such as mechanical parts, organs or monuments. A central question is that of geometric sampling and the transition from the continuous to the discrete. The question of sampling - which has a long history in signal processing - and images - after the seminal work of Claude Shannon in the 1950s - is much more recent in geometry and requires new theoretical tools.

This lecture looks at the case of surfaces, which are used in a wide range of applications in the visualization of three-dimensional objects. Meshing a surface consists in sampling the surface and connecting these points to form a triangulated surface in 3-dimensional space. First of all, we need to characterize a good sample of a surface: we understand that, to offer guarantees, the sample must be sufficiently dense, and the more complex the shape we want to mesh, the denser it must be. We then need to specify the approximation criteria we expect from a mesh, both geometric and topological. All that remains is to build a good mesh. We show that we can adapt Delaunay's notion of triangulation and build a surface meshing algorithm that offers theoretical guarantees and behaves very well in practice.