Chapter 7Geometric Modeling: Methods
In this chapter, we will look at different ways of modeling curves and surfaces based on the nature of their definition, analytical or discrete (set of vertices, edges, faces, etc., hence a triangulation) or, again, completely discrete (a point cloud) and we will also show how to extract the characteristic lines and points of a surface.
In the case where we have an analytical definition, modeling consists of constructing a discrete geometric support (faithful to the geometry), that is an assembly of edges for a curve or a triangulation for a surface. This support will be used instead of the analytical definition, thus simplified and replaced in a unified manner. Let us note that this trick, which has been used for a long time in visualization algorithms, will help us to quickly place points on the curve or surface during the meshing or adaptive remeshing process for said curve or surface.
In the case where the curve or the surface is defined based on an assembly of edges or a triangulation, modeling consists of constructing a geometric support of continuity that is at least of the order 1 (continuous tangents and tangent planes). In other words, contrary to the analytical case, here we reinvent the geometry while remaining faithful to its initial discrete definition. Similarly, this support will be used in meshing or adaptive remeshing problems.
Finally, when the curve or surface is defined based on a point cloud, the modeling consists of ...