High-quality tree structures modelling using local convolution surface approximation

In this paper, we propose a local convolution surface approximation approach for quickly modelling tree structures with pleasing visual effect. Using our proposed local convolution surface approximation, we present a tree modelling scheme to create the structure of a tree with a single high-quality quad-only mesh. Through combining the strengths of the convolution surfaces, subdivision surfaces and GPU, our tree modelling approach achieves high efficiency and good mesh quality. With our method, we first extract the line skeletons of given tree models by contracting the meshes with the Laplace operator. Then we approximate the original tree mesh with a convolution surface based on the extracted skeletons. Next, we tessellate the tree trunks represented by convolution surfaces into quad-only subdivision surfaces with good edge flow along the skeletal directions. We implement the most time-consuming subdivision and convolution approximation on the GPU with CUDA, and demonstrate applications of our proposed approach in branch editing and tree composition

Geometric Structures on Matrix-valued Subdivision Schemes

Surface subdivision schemes are used in computer graphics to generate visually smooth surfaces of arbitrary topology. Applications in computer graphics utilize surface normals and curvature. In this paper, formulas are obtained for the first and second partial derivatives of limit surfaces formed using 1-ring subdivision schemes that have 2 by 2 matrix-valued masks. Consequently, surface normals, and Gaussian and mean curvatures can be derived. Both quadrilateral and triangular schemes are considered and for each scheme both interpolatory and approximating schemes are examined. In each case, we look at both extraordinary and regular vertices. Every 3-D vertex of the refinement polyhedrons also has what is called a corresponding “shape vertex.” The partial derivative formulas consist of linear combinations of surrounding polyhedron vertices as well as their corresponding shape vertices. We are able to derive detailed information on the matrix-valued masks and about the left eigenvectors of the (regular) subdivision matrix. Local parameterizations are done using these left eigenvectors and final formulas for partial derivatives are obtained after we secure detailed information about right eigenvectors of the subdivision matrix. Using specific subdivision schemes, unit normals so obtained are displayed. Also, formulas for initial shape vertices are postulated using discrete unit normals to our original polyhedron. These formulas are tested for reasonableness on surfaces using specific subdivision schemes. Obtaining a specified unit normal at a surface point is examined by changing only these shape vertices. We then describe two applications involving surface normals in the field of computer graphics that can use our results

Subdivision Shell Elements with Anisotropic Growth

A thin shell finite element approach based on Loop's subdivision surfaces is proposed, capable of dealing with large deformations and anisotropic growth. To this end, the Kirchhoff-Love theory of thin shells is derived and extended to allow for arbitrary in-plane growth. The simplicity and computational efficiency of the subdivision thin shell elements is outstanding, which is demonstrated on a few standard loading benchmarks. With this powerful tool at hand, we demonstrate the broad range of possible applications by numerical solution of several growth scenarios, ranging from the uniform growth of a sphere, to boundary instabilities induced by large anisotropic growth. Finally, it is shown that the problem of a slowly and uniformly growing sheet confined in a fixed hollow sphere is equivalent to the inverse process where a sheet of fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless, quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl

Smooth Subdivision Surfaces: Mesh Blending and Local Interpolation

Subdivision surfaces are widely used in computer graphics and animation. Catmull-Clark subdivision (CCS) is one of the most popular subdivision schemes. It is capable of modeling and representing complex shape of arbitrary topology. Polar surface, working on a triangle-quad mixed mesh structure, is proposed to solve the inherent ripple problem of Catmull-Clark subdivision surface (CCSS). CCSS is known to be C1 continuous at extraordinary points. In this work, we present a G2 scheme at CCS extraordinary points. The work is done by revising CCS subdivision step with Extraordinary-Points-Avoidance model together with mesh blending technique which selects guiding control points from a set of regular sub-meshes (named dominative control meshes) iteratively at each subdivision level. A similar mesh blending technique is applied to Polar extraordinary faces of Polar surface as well. Both CCS and Polar subdivision schemes are approximating. Traditionally, one can obtain a CCS limit surface to interpolate given data mesh by iteratively solving a global linear system. In this work, we present a universal interpolating scheme for all quad subdivision surfaces, called Bezier Crust. Bezier Crust is a specially selected bi-quintic Bezier surface patch. With Bezier Crust, one can obtain a high quality interpolating surface on CCSS by parametrically adding CCSS and Bezier Crust. We also show that with a triangle/quad conversion process one can apply Bezier Crust on Polar surfaces as well. We further show that Bezier Crust can be used to generate hollowed 3D objects for applications in rapid prototyping. An alternative interpolating approach specifically designed for CCSS is developed. This new scheme, called One-Step Bi-cubic Interpolation, uses bicubic patches only. With lower degree polynomial, this scheme is appropriate for interpolating large-scale data sets. In sum, this work presents our research on improving surface smoothness at extraordinary points of both CCS and Polar surfaces and present two local interpolating approaches on approximating subdivision schemes. All examples included in this work show that the results of our research works on subdivision surfaces are of high quality and appropriate for high precision engineering and graphics usage

Approximation of Subdivision Surfaces for Interactive Applications

International audienceIn this sketch, we propose a visually plausible approximation subdivision surfaces for interactive applications. The complete idea is discussed in the full paper "QAS: Realtime Quadratic Approximation of Subdivision Surfaces", published in the proceedings of Pacific Graphics 2007 and available online (http://iparla.labri.fr/publications/2007/BS07c/)

Selected applications of subdivision surfaces and numerical quadratures for Gregory patches

Ontwikkelingen in de meetkundige modellering en bewerking hebben eensleutelrol gespeeld in verschillende toepassingen, waarondercomputerspellen, computerondersteund ontwerp envectorgrafiek. Oppervlakken van een willekeurigecomplexiteit zijn onmisbaar op deze gebieden, omdat ze zorgen voor eenflexibele weergave van verschillende vormen.Gebruikmakend van het concept van kenmerk-adaptieve verfijningontwerpen we een nieuwe en flexibele weergave van vectorgrafiek. Deontwerpen kunnen waar nodig lokaal verfijnd worden zodat deresulterende weergave een lagere dichtheid heeft en efficiënter op hetscherm is weer te geven. Onze tweede toepassing verbetert de visuelekwaliteit van vormen door gebruik te maken van een speciale constructiein de buurt van plooien van de gemodelleerde objecten.Onze laatste bijdrage richt zich op een bepaald type oppervlakken diegebruikt zijn in de modelleringscontext, maar waarvan het gebruik innumerieke simulaties tot nu toe beperkt is geweest, grotendeelsvanwege het gebrek aan efficiënte en nauwkeurige evaluatieregels voorintegralen. We vullen deze lacune op door het afleiden vandergelijke regels voor deze oppervlakken.Advancements in geometric modelling and processing have played a key role in various applications, including computer games, computer-aided design, and vector graphics. Surfaces of arbitrary complexity are indispensable in these areas, as they provide a flexible representation of various shapes.Leveraging the concept of feature-adaptive refinement, we design a new and flexible vector graphics representation. The designs can be refined locally where needed so that the resulting representation is less dense and more efficient to render to screen. Our second application enhances the visual appearance of shapes by using a special construction near creases of the modelled objects.Our final contribution focuses on a certain type of surfaces that have been used in the modelling context, but their use in numerical simulations has been limited so far, largely owing to the lack of efficient and accurate integral evaluation rules. We fill this gap by producing such rules for these surfaces