Catmull-Clark Research Articles

细分曲面奇异点处的光滑过渡

首先通过引入形状调节参数c(0≤c≤1)给出了一种形状可调的C-C细分方法，基于该理论，针对细分曲面中的奇异点，本文以奇异点处的2-环作为控制网格，采用循环映射的方法，提出了一种形状可调的G2曲面造型方法，得到Bezier控制点的显式解。与以往的方法相比，生成的曲面不仅在奇异点处达到G2连续，而且解决了曲面设计的可调性问题。本文给出了算法流程和相关数据结构，也给出了相应的实例进行验证。 This paper discusses G2 smooth connection near singular points on subdivision surfaces. A shape adjustable Catmull-Clark subdivision algorithm is present by introducing subdivision shape ad-justment parameter c(0≤c≤1). Based on this, aimed at the singular points in subdivision surface, based on the 2-ring of the singular points as control mesh, adopting the method of cyclic mapping, this paper proposes a G2 surface modeling method which is shape-adjustable. Then we can get the explicit solution of the Bezier control points. Compared with existing methods, the generated surface not only achieves the G2 continuous at singular points, but solves the problem of the curved surface design to be adjustable. The algorithm process and data structure are present. Corresponding examples are also given in this paper.

Isogeometric shape optimisation of shell structures using multiresolution subdivision surfaces

We introduce the isogeometric shape optimisation of thin shell structures using subdivision surfaces. Both triangular Loop and quadrilateral Catmull-Clark subdivision schemes are considered for geometry modelling and finite element analysis. A gradient-based shape optimisation technique is implemented to minimise compliance, i.e. to maximise stiffness. Different control meshes describing the same surface are used for geometry representation, optimisation and finite element analysis. The finite element analysis is performed with subdivision basis functions corresponding to a sufficiently fine control mesh. During iterative shape optimisation the geometry is updated starting from the coarsest control mesh and proceeding to increasingly finer control meshes. The proposed approach is applied to three optimisation examples, namely a catenary, a roof over a rectangular domain, and free-form architectural shell roof. The influence of the geometry description and the used subdivision scheme on the obtained optimised curved geometries are investigated in detail.

Integration of generalized B-spline functions on Catmull–Clark surfaces at singularities

Subdivision surfaces are a common tool in geometric modelling, especially in computer graphics and computer animation. Nowadays, this concept has become established in engineering too. The focus here is on quadrilateral control grids and generalized B-spline surfaces of Catmull–Clark subdivision type. In the classical theory, a subdivision surface is defined as the limit of the repetitive application of subdivision rules to the control grid. Based on Stam’s idea, the labour-intensive process can be avoided by using a natural parameterization of the limit surface. However, the simplification is not free of defects. At singularities, the smoothness of the classically defined limit surface has been lost. This paper describes how to rescue the parameterization by using a subdivision basis function that is consistent with the classical definition, but is expensive to compute. Based on this, we introduce a characteristic subdivision finite element and use it to discretize integrals on subdivision surfaces. We show that in the integral representation the complicated parameterization reduces to a decisive factor. We compare the natural and the characteristic subdivision finite element approach solving PDEs on surfaces. As model problem we consider the mean curvature flow, whereby the computation is done on the step-by-step changing geometry.

G2Interpolation for polar surfaces

ABSTRACTIn this paper, we present a G2 interpolating scheme for Polar surfaces, so that polar surfaces can be used in high precision CAD/CAM applications as well. The new scheme is Bezier crust based, i.e., the interpolating surface is generated by parametrically attaching an especially selected bi-degree 5 Bezier surface to a Catmull-Clark subdivision surface. While Bezier crust based scheme handles quad faces only, we show that through a conversion process, we can handle triangular faces in the Polar part as well. Curvature continuity of the generated limit surface of our new scheme is consistent with the corresponding Polar surface. In case of a PCCS, the generated limit surface of our new scheme is G2 on the Polar part.

Subdividing barycentric coordinates

Barycentric coordinates are commonly used to represent a point inside a polygon as an affine combination of the polygon's vertices and to interpolate data given at these vertices. While unique for triangles, various generalizations to arbitrary simple polygons exist, each satisfying a different set of properties. Some of these generalized barycentric coordinates do not have a closed form and can only be approximated by piecewise linear functions. In this paper we show that subdivision can be used to refine these piecewise linear functions without losing the key barycentric properties. For a wide range of subdivision schemes, this generates a sequence of piecewise linear coordinates which converges to non-negative and C1 continuous coordinates in the limit. The power of the described approach comes from the possibility of evaluating the C1 limit coordinates and their derivatives directly. We support our theoretical results with several examples, where we use Loop or Catmull–Clark subdivision to generate C1 coordinates, which inherit the favourable shape properties of harmonic coordinates or the small support of local barycentric coordinates.

A free shape 3d modeling system for creative design based on modified catmull-clark subdivision

How to express the designer's creative intent in a simple and intuitive way is the main problem in 3d modeling, especially for novice designers. This paper presents a free shape 3d modeling system for creative design based on modified Catmull-Clark subdivision. The system contains a series of easy but novel operations which can be used to change the topology of models, such as creating holes and handles. In order to create sharp features, feature marking operations are provided to specify where the sharp feature is. This system also provides surface conversion function to make the modeling results be compatible with the traditional CAD systems. Firstly, a simple but efficient quad domain division scheme is adopted to generate quad sub-meshes. In order to improve the smoothness at the regular vertices, long boundary curves which across multiple sub-meshes are used to be the boundary constraints while fitting. In this way, the smoothness at regular vertex can be C2 continuous. We perform experiments for both skilled and novice designers. Results show that our system is easy to operate and can be used to construct complex models with less time.

On numerical integration in isogeometric subdivision methods for PDEs on surfaces

Subdivision surfaces offer great flexibility in capturing irregular topologies combined with higher order smoothness. For instance, Loop and Catmull–Clark subdivision schemes provide C2 smoothness everywhere except at extraordinary vertices, where the generated surfaces are C1 smooth. The combination of flexibility and smoothness leads to their frequent use in geometric modeling and makes them an ideal candidate for performing isogeometric analysis of higher order problems with irregular topologies, e.g. thin shell problems. In the neighborhood of extraordinary vertices, however, the resulting surfaces (and the functions defined on them) are non-polynomial. Thus numerical integration based on quadrature has to be performed carefully to preserve the expected higher order consistency.This paper presents a detailed case study of different quadrature schemes for isogeometric discretizations of partial differential equations on closed surfaces with Loop’s subdivision scheme. As model problems we consider elliptic equations with the Laplace–Beltrami and the surface bi-Laplacian operator as well as the associated eigenvalue problem for the Laplace–Beltrami operator. A particular emphasis is on the robustness of the approach in the vicinity of extraordinary vertices. Based on a series of numerical experiments, different quadrature schemes are compared. Mid-edge quadrature, which can easily be implemented via a lookup table, turns out to be a preferable choice due to its robustness and efficiency.

NURBS曲面片的混合与填充

本文提出用轮廓删除法对NURBS曲面片进行细分的曲面混合及N边洞填充的方法。该方法先根据非均匀Catmull-Clark细分原理进行曲面混合，构造插值于角点的非均匀Catmull-Clark细分曲面法，再运用轮廓删除法在细分迭代的过程中去除控制网格边界轮廓，然后进行N边洞的填充以生成一张整体光滑的连续曲面。该方法在保证混合曲面与基曲面片在边界处C2连续时，能使N边洞填充曲面具有较好的光顺性。实例结果表明，该方法简化了具体的计算机实现过程，拓宽了细分曲面应用范围，解决了细分曲面与经典样条拼接的不相容问题，且使生成的曲面兼顾了二者的优点，具有较好的填充效果。 This paper presents a method for subdividing the NURBS surface patches blending and N-sided hole filling by contour deletion method. First, surface blended according to the principle of non- uniform Catmull-Clark subdivision surface, and the non-uniform Catmull-Clark subdivision method which builds the surface interpolating corner vertices and boundary curves was constructed. Then the contour deletion method was used to remove the control mesh boundary contour in the process of segmentation iteration. Last, N-sided hole filling was used to generate an integral smooth continuous surface. The method not only can guarantee the blending surface and the base surface patches C2 continuity at the boundary, but also can make the N-sided hole filling surface have better smoothness. Results show that, this method simplifies the specific computer realization process, broads the scope of application of subdivision surfaces, and solves the problem of incompatibility between subdivision surface and classic spline splicing; the resulting surface has both advantages with better filling effect.

Isogeometric analysis based on extended Catmull–Clark subdivision

In this paper, we propose a subdivision-based finite element method as an integration of the isogeometric analysis (IGA) framework which adopts the uniform representation for geometric modeling and finite element simulation. The finite element function space is induced from the limit form of Catmull–Clark surface subdivision containing boundary subdivision schemes which has C1 continuity everywhere. It is capable of exactly representing complex geometries with any shaped boundaries which are represented as piecewise cubic B-spline curves. It is compatible with modern Computer Aided Design (CAD) software systems. The advantage of this strategy admits quadrilateral meshes of arbitrary topology. In this work, the computational domains with planar geometries are considered. We establish the approximation properties of Catmull–Clark surface subdivision function based on the Bramble–Hilbert lemma. Numerical tests are performed through three Poisson’s equations with the Dirichlet boundary condition to corroborate the theoretical proof.

Extended Truncated Hierarchical Catmull–Clark Subdivision

Abstract In this paper we present an extended Truncated Hierarchical Catmull–Clark Subdivision (eTHCCS) method, which improves the efficiency of local refinement in Truncated Hierarchical Catmull–Clark Subdivision (THCCS). Based on the Stam’s method, we first build a set of basis functions over arbitrary quadrilateral meshes and apply them to THCCS. Then, a new basis-function-insertion scheme is developed with the aid of the truncation mechanism, which refines one-ring neighboring elements rather than two-ring neighborhoods. Therefore, eTHCCS significantly improves the efficiency of local refinement compared with THCCS, as demonstrated by one benchmark problem and several complex models. Moreover, eTHCCS is also proved to preserve the input geometry and produce nested spaces.

On the interpolation of non-iso-parametric curves

AbstractExact curve interpolation has frequently been addressed in a variety of modeling domains. The suggested solutions uniformly assume the interpolated curve to be iso-parametric along one or the other of the parameter lines controlling the interpolating surface. However, once this assumption is abandoned, the difficulty of this problem rises to a degree that no known method is currently in existence which is able to solve this problem efficiently enough for practical purposes. In this context, this paper presents an algorithm capable of embedding a cubic B-spline curve within a regular quad-based Catmull-Clark subdivision surface along any arbitrary direction not necessarily along any of the parameter lines controlling the surface.

Dyadic T-mesh subdivision

Meshes with T-joints (T-meshes) and related high-order surfaces have many advantages in situations where flexible local refinement is needed. At the same time, designing subdivision rules and bases for T-meshes is much more difficult, and fewer options are available. For common geometric modeling tasks it is desirable to retain the simplicity and flexibility of commonly used subdivision surfaces, and extend them to handle T-meshes. We propose a subdivision scheme extending Catmull-Clark and NURSS to a special class of quad T-meshes, dyadic T-meshes, which have no more than one T-joint per edge. Our scheme is based on a factorization with the same structure as Catmull-Clark subdivision. On regular T-meshes it is a refinement scheme for a subset of standard T-splines. While we use more variations of subdivision masks compared to Catmull-Clark and NURSS, the minimal size of the stencil is maintained, and all variations in formulas are due to simple changes in coefficients.

A Heuristic Offsetting Scheme for Catmull-Clark Subdivision Surfaces

ABSTRACTIn rapid prototyping, a hollowed prototype is preferred and significantly reduces the building time and material consumption in contrast to a solid model. Most rapid prototyping obtains solid thin shell by gradually adding or solidifying materials layer by layer. This is a non-trivial problem to offset a solid which involves finding all self-intersections and filling gaps after raw offsetting. While Catmull-Clark subdivision (CCS) surfaces are widely used in solid modeling, the hollow solid/thin shell problems are not well addressed yet. In this paper, we explore earlier methods of obtaining thin shell CCS solid and present a new thin solid approach. With this new scheme, one can efficiently avoid creases and handle gaps.The new scheme is heuristic, but inner surface is parametric, so computation of the inner surface is simplified. And with offsetting Bezier crust applied, the inner surface maintains the mesh structure and continuity of the outer surface. The obtained thin shell solid is continuou...

Truncated hierarchical Catmull–Clark subdivision with local refinement

Abstract In this paper we present a new method termed Truncated Hierarchical Catmull–Clark Subdivision (THCCS), which generalizes truncated hierarchical B-splines to control grids of arbitrary topology. THCCS basis functions satisfy partition of unity, are linearly independent, and are locally refinable. THCCS also preserves geometry during adaptive h -refinement and thus inherits the surface continuity of Catmull–Clark subdivision, namely C 2 -continuous everywhere except at the local region surrounding extraordinary nodes, where the surface continuity is C 1 . Adaptive isogeometric analysis is performed with THCCS basis functions on a benchmark problem with extraordinary nodes. Local refinement on complex surfaces is also studied to show potential wide application of the proposed method.

Skinning cubic Bézier splines and Catmull-Clark subdivision surfaces

Smooth space deformation has become a vital tool for the animation and design of 2D and 3D shapes. Linear methods, under the umbrella term of "linear blend skinning", are the de facto standard for 3D animations. Unfortunately such approaches do not trivially extend to deforming vector graphics , such as the cubic Bézier splines prevalent in 2D or subdivision surfaces in 3D. We propose a variational approach to reposition the control points of cubic Bézier splines and Catmull-Clark subdivision surfaces---or any linear subdivision curves or surfaces---to produce curves or surfaces which match a linear blend skinning deformation as closely as possible. Exploiting the linearity of linear blend skinning, we show how this optimization collapses neatly into the repeated multiplication of a matrix per handle. We support C 0 , C 1 , G 1 , and fixed-angle continuity constraints between adjacent Bézier curves in a spline. Complexity scales linearly with respect to the number of input curves and run-time performance is fast enough for real-time editing and animation of high-resolution shapes.

Local Painting and Deformation of Meshes on the GPU

AbstractWe present a novel method to adaptively apply modifications to scene data stored in GPU memory. Such modifications may include interactive painting and sculpting operations in an authoring tool, or deformations resulting from collisions between scene objects detected by a physics engine. We only allocate GPU memory for the faces affected by these modifications to store fine‐scale colour or displacement values. This requires dynamic GPU memory management in order to assign and adaptively apply edits to individual faces at runtime. We present such a memory management technique based on a scan‐operation that is efficiently parallelizable. Since our approach runs entirely on the GPU, we avoid costly CPU–GPU memory transfer and eliminate typical bandwidth limitations. This minimizes runtime overhead to under a millisecond and makes our method ideally suited to many real‐time applications such as video games and interactive authoring tools. In addition, our algorithm significantly reduces storage requirements and allows for much higher resolution content compared to traditional global texturing approaches. Our technique can be applied to various mesh representations, including Catmull–Clark subdivision surfaces, as well as standard triangle and quad meshes. In this paper, we demonstrate several scenarios for these mesh types where our algorithm enables adaptive mesh refinement, local surface deformations and interactive on‐mesh painting and sculpting.

Computing smooth surface contours with accurate topology

This article introduces a method for accurately computing the visible contours of a smooth 3D surface for stylization. This is a surprisingly difficult problem, and previous methods are prone to topological errors, such as gaps in the outline. Our approach is to generate, for each viewpoint, a new triangle mesh with contours that are topologically equivalent and geometrically close to those of the original smooth surface. The contours of the mesh can then be rendered with exact visibility. The core of the approach is Contour Consistency, a way to prove topological equivalence between the contours of two surfaces. Producing a surface tessellation that satisfies this property is itself challenging; to this end, we introduce a type of triangle that ensures consistency at the contour. We then introduce an iterative mesh generation procedure, based on these ideas. This procedure does not fully guarantee consistency, but errors are not noticeable in our experiments. Our algorithm can operate on any smooth input surface representation; we use Catmull-Clark subdivision surfaces in our implementation. We demonstrate results computing contours of complex 3D objects, on which our method eliminates the contour artifacts of other methods.

A generalized surface subdivision scheme of arbitrary order with a tension parameter

This article presents a generalized B-spline surface subdivision scheme of arbitrary order with a tension parameter. We first propose a tensor-product subdivision scheme that produces ku×kv order generalized B-spline limit surfaces. Generalized B-spline surface is the unified and extended form of B-splines, trigonometric B-splines and hyperbolic B-splines (Fang et al. 2010). The tensor product subdivision scheme can be used to generate various surfaces of revolution, including those generated by classical analytic curves that can be exactly represented by generalized B-spline curves. By extending a bi-order (say k) tensor-product scheme to meshes of arbitrary topology, we further propose a generalized surface subdivision scheme with a tension parameter. Several well-known subdivision schemes, including Doo–Sabin subdivision, Catmull–Clark subdivision, and two other subdivision schemes proposed by Morin et al. (2001) and Stam (2001), become special cases of the generalized subdivision scheme. The tension parameter can be used to adjust the shape of subdivision surfaces. The scheme produces higher order Ck−2 continuous limit surfaces except at extraordinary points where the continuity is C1. Convenient and hierarchical methods are also presented for embedding sharp features and semi-sharp features on the resulting limit surfaces.

A Deformable Generic 3D Model of Haptoral Anchor of Monogenean

In this paper, a digital 3D model which allows for visualisation in three dimensions and interactive manipulation is explored as a tool to help us understand the structural morphology and elucidate the functions of morphological structures of fragile microorganisms which defy live studies. We developed a deformable generic 3D model of haptoral anchor of dactylogyridean monogeneans that can subsequently be deformed into different desired anchor shapes by using direct manipulation deformation technique. We used point primitives to construct the rectangular building blocks to develop our deformable 3D model. Point primitives are manually marked on a 2D illustration of an anchor on a Cartesian graph paper and a set of Cartesian coordinates for each point primitive is manually extracted from the graph paper. A Python script is then written in Blender to construct 3D rectangular building blocks based on the Cartesian coordinates. The rectangular building blocks are stacked on top or by the side of each other following their respective Cartesian coordinates of point primitive. More point primitives are added at the sites in the 3D model where more structural variations are likely to occur, in order to generate complex anchor structures. We used Catmull-Clark subdivision surface modifier to smoothen the surface and edge of the generic 3D model to obtain a smoother and more natural 3D shape and antialiasing option to reduce the jagged edges of the 3D model. This deformable generic 3D model can be deformed into different desired 3D anchor shapes through direct manipulation deformation technique by aligning the vertices (pilot points) of the newly developed deformable generic 3D model onto the 2D illustrations of the desired shapes and moving the vertices until the desire 3D shapes are formed. In this generic 3D model all the vertices present are deployed for displacement during deformation.

A unified method for hybrid subdivision surface design using geometric partial differential equations

Surface subdivision gains popularity in surface design owing to its flexible applications for geometry with complicated topology and simple computational scheme, and the geometric partial differential equation (GPDE) method is an advanced technology for constructing high-quality smooth surfaces. In this paper, we composite these two ingredients to form a unified method for freeform surface design. We choose the mean curvature flow and Willmore flow as our driven GPDEs, and the finite element method coupled with a hybrid Loop and Catmull–Clark subdivision algorithm as the numerical simulation method. This research presents a novel technique to evaluate the finite element basis functions and the first attempt for constructing the GPDE subdivision surface with hybrid control meshes consisting of triangles and quadrilaterals. Numerical experiments show that the construction method is efficient and robust, yielding high-quality hybrid subdivision surfaces.