We present a new globally smooth parameterization method for triangulated surfaces of arbitrary topology. Given two orthogonal piecewise linear vector elds dened over the input mesh (typically the estimated principal curvature directions), our method computes two piecewise linear periodic functions, aligned with the input vector elds, by minimizing an objective function. The bivariate function they dene is a smooth parameterization almost everywhere on the surface, except in the vicinity of singular vertices, edges and triangles, where the derivatives of the parameterization vanish. We extract a quadrilateral chart layout from the parameterization function and propose an automatic procedure to detect the singularities, and x them by splitting and re-parameterizing the containing charts. Our method can construct both quasi-conformal (angle preserving) and quasi-isometric (angle and area preserving) parameterizations. The more restrictive class of quasi-isometric parameterizations is constructed at the expense of introducing more singularities. The constructed parameterizations can be used for a variety of geometry processing applications. Since we can align the parameterization with the principal curvature directions, our result is particularly suitable for surface tting and remeshing.

Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction elds (unit tangent vector elds) dened over a surface. For instance, these direction elds are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such direction elds can be designed in fundamentally dierent 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 unied formalism is still lacking for the mathematical object that characterizes these generalized direction elds. As a consequence, existing direction eld design algorithms are limited to use non-optimum local relaxation procedures.In this paper, we formalize N-symmetry direction elds, a generalization of classical direction elds. We give a new denition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the Poincaré-Hopf theorem in the case of N-symmetry direction elds on 2-manifolds. Based on this theorem, we explain how to control the topology of N-symmetry direction elds on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly ecient algorithm to design a smooth eld interpolating user dened singularities and directions.

A Texture Atlas is an efficient color representation for 3D Paint Systems. The model to be textured is decomposed into charts homeomorphic to discs, each chart is parameterized, and the unfolded charts are packed in texture space. Existing texture atlas methods for triangulated surfaces suffer from several limitations, requiring them to generate a large number of small charts with simple borders. The discontinuities between the charts cause artifacts, and make it difficult to paint large areas with regular patterns.In this paper, our main contribution is a new quasi-conformal parameterization method, based on a least-squares approximation of the Cauchy-Riemann equations. The so-defined objective function minimizes angle deformations, and we prove the following properties: the minimum is unique, independent of a similarity in texture space, independent of the resolution of the mesh and cannot generate triangle flips. The function is numerically well behaved and can therefore be very efficiently minimized. Our approach is robust, and can parameterize large charts with complex borders.We also introduce segmentation methods to decompose the model into charts with natural shapes, and a new packing algorithm to gather them in texture space. We demonstrate our approach applied to paint both scanned and modeled data sets.The remainder of this section presents the existing methods for these three steps, and their limitations with respect to the requirements mentioned above. We then introduce a new texture atlas generation method, meeting these requirements by creating charts with natural shapes, thus reducing texture artifacts.

Many algorithms in texture synthesis, nonphotorealistic rendering (hatching), or remeshing require to define the orientation of some features (texture, hatches, or edges) at each point of a surface. In early works, tangent vector (or tensor) fields were used to define the orientation of these features. Extrapolating and smoothing such fields is usually performed by minimizing an energy composed of a smoothness term and of a data fitting term. More recently, dedicated structures (N -RoSy and N -symmetry direction fields ) were introduced in order to unify the manipulation of these fields, and provide control over the field's topology (singularities). On the one hand, controlling the topology makes it possible to have few singularities, even in the presence of high frequencies (fine details) in the surface geometry. On the other hand, the user has to explicitly specify all singularities, which can be a tedious task. It would be better to let them emerge naturally from the direction extrapolation and smoothing.This article introduces an intermediate representation that still allows the intuitive design operations such as smoothing and directional constraints, but restates the objective function in a way that avoids the singularities yielded by smaller geometric details. The resulting design tool is intuitive, simple, and allows to create fields with simple topology, even in the presence of high geometric frequencies. The generated field can be used to steer global parameterization methods (e.g., QuadCover).

Figure 1: Our algorithm produces smooth frame fields in volumes. Frames (a) are represented by spherical harmonic functions (b), attached to each vertex of a tetrahedral mesh. Streamlines and singularities of the field are shown in yellow and red, respectively.

Surface materials are commonly described by attributes stored in textures (for instance, color, normal, or displacement). Interpolation during texture lookup provides a continuous value field everywhere on the surface, except at the chart boundaries where visible discontinuities appear. We propose a solution to make these seams invisible, while still outputting a standard texture atlas. Our method relies on recent advances in quad remeshing using global parameterization to produce a set of texture coordinates aligning texel grids across chart boundaries. This property makes it possible to ensure that the interpolated value fields on both sides of a chart boundary precisely match, making all seams invisible. However, this requirement on the uv coordinates needs to be complemented by a set of constraints on the colors stored in the texels. We propose an algorithm solving for all the necessary constraints between texel values, including through different magnification modes (nearest, bilinear, biquadratic and bicubic), and across facets using different texture resolutions. In the typical case of bilinear magnification and uniform resolution, none of the texels appearing on the surface are constrained. Our approach also ensures perfect continuity across several MIP-mapping levels.

Figure 1: Our algorithm traces polylines on triangulated surfaces. Unlike streamline tracing algorithms, polylines produced by our technique cannot cross each others. It works even with highly perturbed surfaces (left) and supports any type of vector field singularities (right). This property is required to segment surfaces with chart boundaries aligned with a vector field (right). AbstractWe propose an algorithm for tracing polylines on a triangle mesh such that: they are aligned with a Nsymmetry direction field, and two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is very difficult to enforce with numerical integration of vector fields. We propose an alternative solution based on "stream-mesh", a new combinatorial data structure that defines, for each point of a triangle edge, where the corresponding polyline leaves the triangle. It makes it possible to trace polylines by iteratively crossing trian- * e-mail: ray@loria.fr † e-mail: sokolovd@loria.fr gles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled. The polylines inherits the cross-free property of the stream-mesh, except inside triangles where avoiding local overlaps would require higher order polycurves.

Protein docking is a fundamental biological process that links two proteins. This link is typically defined by an interaction between two large zones of the protein boundaries. Visualizing such an interface is useful to understand the process thanks to 3D protein structures, to estimate the quality of docking simulation results, and to classify interactions in order to predict docking affinity between classes of interacting zones. Since the interface may be defined by a surface that separates the two proteins, it is possible to create a map of interaction that allows comparisons to be performed in 2D. This paper presents a very fast algorithm that extracts an interface surface and creates a valid and low-distorted interaction map. Another benefit of our approach is that a pre-computed part of the algorithm enables the surface to be updated in real-time while residues are moved.