A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifolds

We introduce a new manifold-based construction for fitting a smooth surface to a triangle mesh of arbitrary topology. Our construction combines in novel ways most of the best features of previous constructions and, thus, it fills the gap left by them. We also introduce a theoretical framework that provides a sound justification for the correctness of our construction. Finally, we demonstrate the effectiveness of our manifold-based construction with a few concrete examples

A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifolds

We introduce a new manifold-based construction for fitting a smooth surface to a triangle mesh of arbitrary topology. Our construction combines in novel ways most of the best features of previous constructions and, thus, it fills the gap left by them. We also introduce a theoretical framework that provides a sound justification for the correctness of our construction. Finally, we demonstrate the effectiveness of our manifold-based construction with a few concrete examples

Piecewise Rational Manifold Surfaces with Sharp Features

We present a construction of a piecewise rational free-form surface of arbitrary topological genus which may contain sharp features: creases, corners or cusps. The surface is automatically generated from a given closed triangular mesh. Some of the edges are tagged as sharp ones, defining the features on the surface. The surface is C s smooth, for an arbitrary value of s, except for the sharp features defined by the user. Our method is based on the manifold construction and follows the blending approach

Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology

We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework does not require manifold constructions and is general enough to allow non-orientable surfaces. We describe compatibility conditions on the transition maps so that the space of differentiable functions is ample and show that these conditions are necessary and sufficient to construct ample spline spaces. We determine the dimension of the space of G1 spline functions which are of degree less than or equal to k on triangular pieces and of bi-degree less than or equal to (k, k) on rectangular pieces, for k big enough. A separability property on the edges is involved to obtain the dimension formula. An explicit construction of basis functions attached resspectively to vertices, edges and faces is proposed; examples of bases of G1 splines of small degree for topological surfaces with boundary and without boundary are detailed

Construction of C\u3csup\u3e∞\u3c/sup\u3e Surfaces From Triangular Meshes Using Parametric Pseudo-Manifolds

We present a new constructive solution for the problem of fitting a smooth surface to a given triangle mesh. Our construction is based on the manifold-based approach pioneered by Grimm and Hughes. The key idea behind this approach is to define a surface by overlapping surface patches via a gluing process, as opposed to stitching them together along their common boundary curves. The manifold based approach has proved to be well-suited to fit with relative ease, Ck-continuous parametric surfaces to triangle and quadrilateral meshes, for any arbitrary finite k or even k = ∞. Smooth surfaces generated by the manifold-based approach share some of the most important properties of splines surfaces, such as local shape control and fixed-sized local support for basis functions. In addition, the differential structure of a manifold provides us with a natural setting for solving equations on the surface boundary of 3D shapes. Our new manifold-based solution possesses most of the best features of previous constructions. In particular, our construction is simple, compact, powerful, and flexible in ways of defining the geometry of the resulting surface. Unlike some of the most recent manifold-based solutions, ours has been devised to work with triangle meshes. These meshes are far more popular than any other kind of mesh encountered in computer graphics and geometry processing applications. We also provide a mathematically sound theoretical framework to undergird our solution. This theoretical framework slightly improves upon the one given by Grimm and Hughes, which was used by most manifold-based constructions introduced before

Flexible G1 Interpolation of Quad Meshes

International audienceTransforming an arbitrary mesh into a smooth G1 surface has been the subject of intensive research works. To get a visual pleasing shape without any imperfection even in the presence of extraordinary mesh vertices is still a challenging problem in particular when interpolation of the mesh vertices is required. We present a new local method, which produces visually smooth shapes while solving the interpolation problem. It consists of combining low degree biquartic Bézier patches with minimum number of pieces per mesh face, assembled together with G1-continuity. All surface control points are given explicitly. The construction is local and free of zero-twists. We further show that within this economical class of surfaces it is however possible to derive a sufficient number of meaningful degrees of freedom so that standard optimization techniques result in high quality surfaces

Point-Normal Subdivision Curves and Surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M

Arbitrary topology meshes in geometric design and vector graphics

Meshes are a powerful means to represent objects and shapes both in 2D and 3D, but the techniques based on meshes can only be used in certain regular settings and restrict their usage. Meshes with an arbitrary topology have many interesting applications in geometric design and (vector) graphics, and can give designers more freedom in designing complex objects. In the first part of the thesis we look at how these meshes can be used in computer aided design to represent objects that consist of multiple regular meshes that are constructed together. Then we extend the B-spline surface technique from the regular setting to work on extraordinary regions in meshes so that multisided B-spline patches are created. In addition, we show how to render multisided objects efficiently, through using the GPU and tessellation. In the second part of the thesis we look at how the gradient mesh vector graphics primitives can be combined with procedural noise functions to create expressive but sparsely defined vector graphic images. We also look at how the gradient mesh can be extended to arbitrary topology variants. Here, we compare existing work with two new formulations of a polygonal gradient mesh. Finally we show how we can turn any image into a vector graphics image in an efficient manner. This vectorisation process automatically extracts important image features and constructs a mesh around it. This automatic pipeline is very efficient and even facilitates interactive image vectorisation