Section: New Results
Topology of vector fields
The quad-remeshing algorithms that we developed last year (  and  ) generate quadrilateral elements aligned with a given guidance vector field. This guidance vector field can be obtained from an estimate of the principal directions of curvature of the surface. However, specific situations may require using alternative vector fields, possibly designed by the user. Moreover, many other algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such direction fields can be designed in fundamentally different ways, according to the symmetry requested: inverting a direction or swapping two directions may be allowed or not.
Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized direction fields. As a consequence, existing direction field design algorithms are constrained to use non-optimum local relaxation procedures.
We developed a formalization of what we call a N -symmetry direction field, a generalization of classical direction fields. We gave a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we wrote an accessible demonstration of the Poincaré-Hopf theorem in the case of N -symmetry direction fields on 2-manifolds. Based on this theorem, we explained how to control the topology of N -symmetry direction fields on meshes. We showed the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user defined singularities and directions [Oops!] .
Visualization of industrial models
As shown in Figure 11 , industrial models can comprise a large number of tubular shapes. This is especially true in the domain of oil and gas, thinking for instance about a tanker, a refinery or an oil platform, which have a very large number of tubes and pipes. Rendering these pipes is traditionally done by first converting them into mesh models (Figure 11 left). This causes jagged silhouettes (a), incorrect intersections (b), and artifacts where the discretizations do not line-up (c). In the frame of his Ph.D. [Oops!] , Rodrigo Toledo developed methods to recover the equation of the geometry, and rendering it directly (Figure 11 right).
To display these surfaces in real-time, we developed new ray-casting algorithms [Oops!] . The ray-casting of implicit surfaces on GPU has been explored in the last few years. However, until recently, they were restricted to second degree (quadrics). We present an iterative solution to ray cast cubics and quartics on GPU. Our solution targets efficient implementation, obtaining interactive rendering for thousands of surfaces per frame. We have given special attention to torus rendering since it is a useful shape for multiple CAD models. We have tested four different iterative methods, including a novel one.
Molecules can be roughly described as the union of spheres (that correspond to the individual atoms). The boundary of this union of spheres is called the Van-der-Waals surface. However, the function of complex molecules depends on more elaborate geometrical objects, deduced from this union of spheres, called molecular surfaces. The most popular definition of molecular surfaces by Connolly models the electric field generated by the interaction of the molecule and a solvent. A Connolly surface is defined to be the set of contact points of a sphere that rolls on the Van-der-Walls surface. More recently, Edelsbrunner introduced the so-called “skin surfaces”  , that are smoother than Connolly surfaces. To visualize a molecular surface, the standard approach consists in triangulating it, and sending the triangles to the graphic board. However, generating the triangulated surface is a time-consuming process. Moreover, depending on the used number of triangles, the quality of the generated image may be unsatisfactory. More importantly, guaranteeing that the topology of the so-constructed triangulated surface corresponds to the initial one faithfully is a difficult theoretic problem  .
For these reasons, our goal is to use the equation of the skin surface directly, and visualize a pixel-accurate version using the GPU (Graphic Processing Unit). Some early results of Matthieu Chavent are shown in Figure 12 -A. The equation of a skin surface is a piecewise quadric surface. The pieces are shown in different colors in Figure 12 -B. The pieces correspond to the cells of a computational geometry data structure, called the mixed complex, that may be thought-of as the linear interpolation between a Delaunay and a Voronoi diagram (see Figure 12 -C).