Bivariate Hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in $\mathbb{R}^3$ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points $\{(x_i, y_i)\}_{i=1}^N$ from which none of the pairs $(x_i,y_i)$ and $(x_j,y_j)$ with $i\neq j$ coincide, it is proved that the resulting surface (function) is $C^1$. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is $C^2$ if the data are not 'too irregular'. Finally the approximation properties of the methods are investigated

A 10-point interpolatory recursive subdivision algorithm for the generation of parametric surfaces

In this paper, an interpolatory subdivision algorithm for surfaces over arbitrary triangulations is introduced and its properties over uniform triangulations studied. The Butterfly Scheme, which is introduced by Dyn, Gregory and Levin is a special case of this algorithm. In our analysis, the matrix approach is employed and the idea of "Cross Difference of Directional Divided Difference" analysis is presented. This method is a generalization of the technique used by Dyn, Gregory and Levin etc. to analyse univariate subdivision algorithms. It is proved that the algorithm produces smooth surfaces provided the shape parameters are kept within an appropriate range

An interpolatory subdivision algorithm for surfaces over arbitrary triangulations

In this paper, an interpolatory subdivision algorithm for surfaces over ar-bitrary triangulations is introduced and its convergence properties over nonuni-form triangulations studied. The so called Butterfly Scheme (interpolatory) is a special case of this algorithm. In our analysis of the algorithm over uniform triangulations, a matrix approach is employed and the idea, of "Cross Differ-ence of Directional Divided Difference" analysis is presented. This method is a generalization of the technique used by Dyn, Gregory and Levin etc. to analyse univariate subdivision algorithms. While for nonuniform data, an extraordi-nary point analysis is introduced and the local subdivision matrix analysis is presented. It is proved that the algorithm produces smooth surfaces over ar-bitrary triangular networks provided the shape parameters are kept within an appropriate range

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

Non-uniform interpolatory subdivision schemes with improved smoothness

Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate C-1 and C-2 limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one. (C) 2022 The Author(s). Published by Elsevier B.V