We present a local refinement approach for gradient meshes, a primitive commonly used in the design of vector illustrations with complex colour propagation. Local refinement allows the artist to add more detail only in the regions where it is needed, as opposed to global refinement which often clutters the workspace with undesired detail and potentially slows down the workflow. Moreover, in contrast to existing implementations of gradient mesh refinement, our approach ensures mathematically exact refinement. Additionally, we introduce a branching feature that allows for a wider range of mesh topologies, as well as a feature that enables sharp colour transitions similar to diffusion curves, which turn the gradient mesh into a more versatile and expressive vector graphics primitive.
We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces based on quad meshes in the context of isogeometric analysis. Starting with a sparse control mesh, the subdivision process generates a sequence of finer and finer quad meshes that in the limit defines a smooth subdivision surface, which can be of any manifold topology. Traditional approaches to quadrature on such surfaces rely on per-quad integration, which is inefficient and typically also inaccurate near vertices where other than four quads meet. Instead, we explore the space of possible groupings of quads and identify the optimal macro-quads in terms of the number of quadrature points needed. We show that macro-quads consisting of quads from one or several consecutive levels of subdivision considerably reduce the cost of numerical integration. Our rules possess a tensor product structure and the underlying univariate rules are Gaussian, i.e., they require the minimum possible number of integration points in both univariate directions. The optimal quad groupings differ depending on the particular application. For instance, computing surface areas, volumes, or solving the Laplace problem lead to different spline spaces with specific structures in terms of degree and continuity. We show that in most cases the optimal groupings are quad-strips consisting of (1 × n) quads, while in some cases a special macro-quad spanning more than one subdivision level offers the most economical integration. Additionally, we extend existing results on exact integration of subdivision splines. This allows us to validate our approach by computing surface areas and volumes with known exact values. We demonstrate on several examples that our quadratures use fewer quadrature points than traditional quadratures. We illustrate our approach to subdivision spline quadrature on the well-known Catmull-Clark scheme based on bicubic splines, but our ideas apply also to subdivision schemes of arbitrary bidegree, including nonuniform and hierarchical variants. Specifically, we address the problems of computing areas and volumes of Catmull-Clark subdivision surfaces, as well as solving the Laplace and Poisson PDEs defined over planar unstructured quadrilateral meshes in the context of isogeometric analysis.
Among the bivariate subdivision schemes available, spline-based schemes, such as Catmull-Clark and Loop, are the most commonly used ones. These schemes have known continuity and can be evaluated at arbitrary parameter values. In this work, we develop a C 1 spline-based scheme based on cubic half-box splines. Although the individual surface patches are triangular, the associated control net is three-valent and thus consists in general of mostly hexagons. In addition to introducing stencils that can be applied in extraordinary regions of the mesh, we also consider boundaries. Moreover, we show that the scheme exhibits ineffective eigenvectors. Finally, we briefly consider architectural geometry and isogeometric analysis as selected applications.
Spiro splines [Lev09] are an interesting alternative for font design and are available in FontFoRge and InKscape. ‡ Two of these attributes are kerning and ligatures, which are both used extensively by X Ǝ T E X. Spotting ligatures in a text is an enjoyable game and often provides a reasonable indication of what a font has to offer! 13 2.1 The above might raise the question whether composite curves connecting with e.g. C 1 continuity can be represented more efficiently, as the middle of these three co-linear points is merely a convex combination of the other two. The answer is positive and can be obtained by studying B-spline curves. 2.1.2 B-spline curves B-splines † come in many different flavours, can be defined using a variety of ways, and form an extensive area of research. Merely interpreting them as an efficient way to represent composite Béziers does perhaps not do them justice, but as this is the direction we come from, it is our starting point. An important aspect to consider when connecting Bézier curves is whether all segments should be defined on parameter intervals of the same length. If we choose to do so, and set that interval to be of unit length, we obtain a vector of parameter values Ξ = [0, 1, 2,. .. , m], where m indicates the number of segments. These parameter values indicate where our segments are connected, or tied together, in parameter space. For this reason, they are often referred to as knots, and Ξ as the knot-vector. Let us consider a composite quadratic Bézier curve of two segments that connect with C 1 continuity. We thus have control points P 0 ,. .. , P 4 , with as corresponding basis functions M 0 (t),. .. , M 4 (t) the Bernsteins, which for the second curve segment are shifted by one unit: M 0 (t) = (1 − t) 2 for t ∈ [0, 1], M 1 (t) = 2t(1 − t) for t ∈ [0, 1], M 2 (t) = t 2 (2 − t) 2 for t ∈ [0, 1], for t ∈ [1, 2], M 3 (t) = 2(t − 1)(2 − t) for t ∈ [1, 2], M 4 (t) = (t − 1) 2 for t ∈ [1, 2].
In this work we develop a design-through-analysis methodology by extending the concept of the NURBS-enhanced finite element method (NE-FEM) to volumes bounded by Catmull-Clark subdivision surfaces. The representation of the boundary as a single watertight manifold facilitates the generation of an external curved triangular mesh, which is subsequently used to generate the interior volumetric mesh. Following the NEFEM framework, the basis functions are defined in the physical space and the numerical integration is realized with a special mapping which takes into account the exact definition of the boundary. Furthermore, an appropriate quadrature strategy is proposed to deal with the integration of elements adjacent to extraordinary vertices (EVs). Both theoretical and practical aspects of the implementation are discussed and are supported with numerical examples.
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