Catmull-Clark Research Articles

The Contrast for Offset Method of Catmull-Clark Subdivision Surface

The offset surface for Catmull-Clark subdivision surface is accomplished in this paper. Currently, the offset methods main two ways, one is directly offset method and the other is inverse iteration offset method. Firstly, this paper summarizes the offset principle for the two offset methods. Secondly, define Catmull-Clark subdivision surface as the research target, the offset surface is obtained by used two offset methods. Lastly, as a case study for molar restorations, the offset error is discussed for two offset methods.

An Efficient Data Structure for Representing Trilateral/Quadrilateral Subdivision Surfaces

Abstract With the increase of subdivision depths, some problems of common data structures for representing the subdivision surfaces appear, such as excessive computer memory consumption and low efficiency of the data query which restrict the popularization and application of subdivision surfaces in more fields. By utilizing the topological characteristics of the subdivision surface, a two-layer data structure named CELL is presented in order to better realize the piecewise representation of trilateral/quadrilateral subdivision surfaces. The inner structure of CELL represents the subdivision surface patch by using arrays, and the outer structure of CELL represents the topological relations between the subdivision surface patches. Based on Catmull-Clark subdivision scheme, the structural compositions of CELL and the realization mechanism of the subdivision algorithm are proposed. Additionally, sharp and semi-sharp features are constructed, and a primary study on amalgamation of the image/Z-map model and subdivision surface is presented. The results of the experimental and theoretical analysis show the superior performance of CELL with relation to computer memory consumption, data query, subdivision surface computation and algorithm development.

Pattern mapping with quad-pattern-coverable quad-meshes

We show that for every surface of positive genus, there exist many quadrilateral manifold meshes that can be texture-mapped with locally translated copies of a single square-texture pattern. This implies, for instance, that every positive-genus surface can be covered seamlessly with any of the 17 plane symmetric wallpaper patterns. We identify sufficient conditions for meshes to be classified as “quad-pattern-coverable”, and we present several methods to construct such meshes. Moreover, we identify some mesh operations that preserve the quad-pattern-coverability property. For instance, since vertex insertion remeshing, which is the remeshing operation behind Catmull–Clark subdivision, preserves quad-pattern-coverability, it is possible to cover any surface of positive genus with iteratively finer versions of the same texture.

Topologically robust B-spline surface reconstruction from point clouds using level set methods and iterative geometric fitting algorithms

In this paper, we present a procedure for automatically reconstructing an arbitrary topological surface from an unorganized point data set; this surface will have three representations, namely quadrilateral meshes, Catmull–Clark subdivision surfaces, and B-spline surfaces. Our novel reconstruction method adapts a level set method to capture the topology of the point clouds in a robust manner and then employs an iterative geometric fitting algorithm to generate high-quality Catmull–Clark subdivision surfaces. A quadrilateral mesh is generated by projecting the control net of the resulting Catmull–Clark surface onto its limit surface. Finally, the control net of the Catmull–Clark surface is converted to that of a B-spline surface. The reconstructed models of topologically complex models show the effectiveness of the proposed algorithm.

An alternative method for constructing interpolatory subdivision from approximating subdivision

This paper presents a new perspective for constructing interpolatory subdivision from primal approximating subdivision. The basic idea is constructing the subdivision rule for new inserted vertices of a new interpolatory subdivision scheme based on an approximating subdivision algorithm applied to a local configuration of the mesh with one vertex updated for interpolation of the vertex. This idea is demonstrated by presenting two new interpolatory subdivision schemes based on Catmull–Clark subdivision for an arbitrary polygonal mesh and Loop subdivision for a triangular mesh, respectively. These algorithms are simple and have a small stencil for computing new points. The new perspective also shows a link between those classic approximating and interpolatory subdivision algorithms such as cubic B-spline curve subdivision and the four-point interpolatory subdivision, Catmull–Clark subdivision and Kobbeltʼs interpolatory scheme, and Loop subdivision and the butterfly algorithm.

Sketch based 3D modeling with curvature classification

In this paper, we introduce a simple method for sketching 3D models in arbitrary topology. Using this method, we have developed a system to convert silhouette sketches to 3D meshes that mostly consists of quadrilaterals and 4-valent vertices. Because of their regular structures, these 3D meshes can effectively be smoothed using Catmull–Clark subdivision. Our method is based on the identification of corresponding points on a set of input curves. Using the structure of correspondences on the curves, we partition curves into junction, cap and tubular regions and construct mostly quadrilateral meshes using this partitioning.

A Quick Edit Method of Curve Network’s Interpolated Surface Based on Combined Subdivision

A simple and effective combined subdivision surface’s modification method is proposed, which is realized by editing its curve network locally. After discussing the overlapping principles of Catmull-Clark subdivision and combined subdivision, a quick update method of the modified surface is given by building the difference curve network from the original curve network and the edited curve network and overlapping the mesh from the difference curve network to the mesh from the original curve network. This method is closely related with the features of combined subdivision, so it is simple and high efficiency. It is useful in the concept design of complex product.

Feature-adaptive GPU rendering of Catmull-Clark subdivision surfaces

We present a novel method for high-performance GPU-based rendering of Catmull-Clark subdivision surfaces. Unlike previous methods, our algorithm computes the true limit surface up to machine precision, and is capable of rendering surfaces that conform to the full RenderMan specification for Catmull-Clark surfaces. Specifically, our algorithm can accommodate base meshes consisting of arbitrary valence vertices and faces, and the surface can contain any number and arrangement of semisharp creases and hierarchically defined detail. We also present a variant of the algorithm which guarantees watertight positions and normals, meaning that even displaced surfaces can be rendered in a crack-free manner. Finally, we describe a view-dependent level-of-detail scheme which adapts to both the depth of subdivision and the patch tessellation density. Though considerably more general, the performance of our algorithm is comparable to the best approximating method, and is considerably faster than Stam's exact method.

Automatic Construction of Quad-Based Subdivision Surfaces Using Fitmaps

We present an automatic method to produce a Catmull-Clark subdivision surface that fits a given input mesh. Its control mesh is coarse and adaptive, and it is obtained by simplifying an initial mesh at high resolution. Simplification occurs progressively via local operators and addresses both quality of surface and faithfulness to the input shape throughout the whole process. The method is robust and performs well on rather complex shapes. Displacement mapping or normal mapping can be applied to approximate the input shape arbitrarily well.

Non-uniform subdivision approach for free-form curves and surfaces of arbitrary degree

In this paper,a new recursive non-uniform subdivision approach for modeling free-form curves and surfaces was presented.Based on the knot insertion technique,the non-uniform subdivision rules were provided for modeling arbitrary-degree curves and surfaces of arbitrary topology with non-uniform knot intervals.In particular,this approach generalized the traditional uniform subdivisions,such as Doo-Sabin and Catmull-Clark subdivision.

A front-tracking method with Catmull–Clark subdivision surfaces for studying liquid capsules enclosed by thin shells in shear flow

This paper presents a front-tracking method for studying the large deformation of a liquid capsule enclosed by a thin shell in a shear flow. The interaction between the fluid and the shell body is accomplished through an implicit immersed boundary method. An improved thin-shell model for computing the forces acting on the shell middle surface during the deformation is described in surface curvilinear coordinates and within the framework of the principle of virtual displacements. This thin-shell model takes full account of in-plane tensions and bending moments developing due to the shell thickness and a preferred three-dimensional membrane structure. The approximation of the shell middle surface is performed through the use of the Catmull–Clark subdivision surfaces. The resulting limit surface is C 2-continuous everywhere except at a small number of extraordinary nodes where it retains C 1 continuity. The smoothness of the limit surface significantly improves the ability of our method in simulating capsules enclosed by hyperelastic thin shells with different shapes and physical properties. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical, ellipsoidal and biconcave capsules in the shear flow. In addition, different types of motion such as tank-treading, swinging, tumbling and transition from tumbling to swinging have been studied over a range of shear rates, viscosity ratios and bending modulus.

Implicit Hierarchical Quad‐Dominant Meshes

AbstractWe present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.

2112 レベルセット法による点群からの位相頑強なメッシュ生成(OS3-2 デジタルエンジニアリングII,OS3 デジタルエンジニアリング)

In this paper, we propose a novel method for approximating a set of scattered data points by a quadrilateral mesh using level set methods followed by geometric algorithms. The level set method is used to capture the topology of the point clouds, and the geometric algorithm using Catmull-Clark subdivision surface is employed to generate high quality meshes. Finally the Catmull-Clark control mesh is converted to B-spline representation. The reconstructed models of topologically complex models show the effectiveness of the proposed algorithm.

Basic Research on Five-Axis NC Machining of Subdivision Surface

Basic research on five-axis NC machining of subdivision surface is presented. Research on Catmull-Clark subdivision surface, a method is presented by using computational geometry techniques to determine optimal tool position and orientation for 5-axis machining with torus end-mill: The normal vector of the subdivision surface vertex and the offset surface calculation are discussed firstly, then the rough machining model of subdivision surface is built and machined by using five-axis machining with the motion of the torus end-mill position and orientation according to the normal line of the surface. The tool position and orientation, cutting radius and tool path are calculated. Finally a computer simulation instantiation is given. This method improves the machining ability of subdivision surfaces.

A new cross subdivision scheme for surface design

In this paper, a new primal approximation 2 -subdivision scheme for surface design, being named the cross scheme or C-scheme for short, is presented. The C 2 -continuity of the C-scheme is mathematically analyzed. The new scheme can be effectively applied to any 3D mesh of quadrilaterals. Extraordinary vertices and miscellaneous boundary scenarios are handled as well. Extensive implementation shows that it performs better than the classical Catmull–Clark subdivision scheme both at nearby vertices and along the non-vertical and non-horizontal directions.

Dinus

The Double Insertion, Nonuniform, Stationary subdivision surface (DINUS) generalizes both the nonuniform, bicubic spline surface and the Catmull-Clark subdivision surface. DINUS allows arbitrary knot intervals on the edges, allows incorporation of special features, and provides limit point as well as limit normal rules. It is the first subdivision scheme that gives the user all this flexibility and at the same time all essential limit information, which is important for applications in modeling and adaptive rendering. DINUS is also amenable to analysis techniques for stationary schemes. We implemented DINUS as an Autodesk Maya plugin to show several modeling and rendering examples.

Compound Biorthogonal Wavelets on Quadrilaterals and Polar Structures

In geometric models with high-valence vertices, current subdivision wavelets may not deal with the special cases well for good visual effect of multiresolution surfaces. In this paper, we present the novel biorthogonal polar subdivision wavelets, which can efficiently perform wavelet analysis to the control nets with polar structures. The polar subdivision can generate more natural subdivision surfaces around the high-valence vertices and avoid the ripples and saddle points where Catmull-Clark subdivision may produce. Based on polar subdivision, our wavelet scheme supports special operations on the polar structures, especially suitable to models with many facets joining. For seamless fusing with Catmull-Clark subdivision wavelet, we construct the wavelets in circular and radial layers of polar structures, so can combine the subdivision wavelets smoothly for composite models formed by quadrilaterals and polar structures. The computations of wavelet analysis and synthesis are highly efficient and fully in-place. The experimental results have confirmed the stability of our proposed approach.

A simple method for interpolating meshes of arbitrary topology by Catmull–Clark surfaces

Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh. The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the given mesh with modified geometric rule. Two methods—push-back operation based method and normal-based method—are presented for the new geometric rule. The interpolation method has the following features: (1) Efficiency: we obtain a generalized cubic B-spline surface to interpolate any given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rule to construct control mesh for the interpolating subdivision surface. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each edge and face, there is one degree of freedom to adjust the shape of the limit surface. These features make interpolation using Catmull–Clark surfaces very simple and thus make the method itself suitable for interactive free-form shape design.

Lens-shaped surfaces and C 2 subdivision

Lens-shaped surfaces (with vertices of valence 2) arise for example in automatic quad-remeshing. Applying standard Catmull–Clark subdivision rules to a vertex of valence 2, however, does not yield a C 1 surface in the limit. When correcting this flaw by adjusting the vertex rule, we discover a variant whose characteristic ring is z → z 2. Since this conformal ring is of degree bi-2 rather than bi-3, it allows constructing a subdivision algorithm that works directly on the control net and generates C 2 limit surfaces of degree bi-4 for lens-shaped surfaces. To further improve shape, a number of re-meshing and re-construction options are discussed indicating that a careful approach pays off. Finally, we point out the analogy between characteristic configurations and the conformal maps z 4/n, cos z and e z.

Parallel smoothing of quad meshes

For use in real-time applications, we present a fast algorithm for converting a quad mesh to a smooth, piecewise polynomial surface on the Graphics Processing Unit (GPU). The surface has well-defined normals everywhere and closely mimics the shape of Catmull–Clark subdivision surfaces. It consists of bicubic splines wherever possible, and a new class of patches—c-patches—where a vertex has a valence different from 4. The algorithm fits well into parallel streams so that meshes with 12,000 input quads, of which 60% have one or more non-4-valent vertices, are converted, evaluated and rendered with 9×9 resolution per quad at 50 frames per second. The GPU computations are ordered so that evaluation avoids pixel dropout.