Subdivision Depth Research Articles

A versatile hierarchical mesh for subdivision surfaces

Most of the needs that occur in applications using subdivision surfaces are covered by our hierarchical mesh data structure. We tested the proposed data structure in some implementations for adaptive visualizations where a highly efficient data structure is essential. Primarily, this means our mesh must support a fast hierarchical refinement of the object with a fast access to the vertices in the different subdivision levels. Only the parts of the object with a contribution to the light transport are computed and stored up to an adequate subdivision depth. A rapid and stable navigation on the mesh must be possible, and storing of ancillary information is required. The modular concept of our mesh enables us to exchange the rules of the subdivision scheme easily. With our hierarchical mesh, the rules for Catmull-Clark, Loop and ESubs, which is another subdivison scheme, were easy to implement.

A bound on the approximation of a Catmull–Clark subdivision surface by its limit mesh

A Catmull–Clark subdivision surface (CCSS) is a smooth surface generated by recursively refining its control meshes, which are often used as linear approximations to the limit surface in geometry processing. For a given control mesh of a CCSS, by pushing the control points to their limit positions, another linear approximation—a limit mesh of the CCSS is obtained. In general a limit mesh might approximate a CCSS better than the corresponding control mesh. We derive a bound on the distance between a CCSS patch and its limit face in terms of the maximum norm of the second order differences of the control points and a constant that depends only on the valence of the patch. A subdivision depth estimation formula for the limit mesh approximation is also proposed. For a given error tolerance, fewer subdivision steps are needed if the refined control mesh is replaced with the corresponding limit mesh.

Improved error estimate for extraordinary Catmull–Clark subdivision surface patches

Based on an optimal estimate of the convergence rate of the second order norm, an improved error estimate for extraordinary Catmull–Clark subdivision surface (CCSS) patches is proposed. If the valence of the extraordinary vertex of an extraordinary CCSS patch is even, a tighter error bound and, consequently, a more precise subdivision depth for a given error tolerance, can be obtained. Furthermore, examples of adaptive subdivision illustrate the practicability of the error estimation approach.

Subdivision Depth Computation for Catmull-Clark Subdivision Surfaces

AbstractA subdivision depth computation technique for Catmull-Clark subdivision surfaces (CCSS’s) is presented. The subdivision depth computation technique also includes distance evaluation techniques for CCSS patches with their control meshes. The distance and the subdivision depth computation techniques provide the long-needed precision/error control tools in subdivision surface trimming, finite element mesh generation, Boolean operations, and surface tessellation for rendering processes.

Computational formula of depth for Catmull–Clark subdivision surfaces

In this paper, by introducing the concept of neighbor points and using the first-order difference of control points of Catmull–Clark surfaces, we obtain the rate of convergence of control meshes of Catmull–Clark surface. By the result of convergence we derive a computational formula of subdivision depth for Catmull–Clark surfaces.

Adaptive Subdivision of Catmull-Clark Subdivision Surfaces

AbstractCatmull-Clark subdivision scheme provides a powerful method for building smooth and complex surfaces. But the number of faces in the uniformly refined meshes increases sharply with respect to subdivision depth. This paper presents an adaptive subdivision technique as a solution to this problem. Instead of subdivision depths of mesh faces, the adaptive subdivision process is driven by labels of mesh vertices, which can be viewed as subdivision depths of the surface in the vicinity of the mesh vertices. Smooth transition between faces with different subdivision depths is provided by an unbalanced-subdivision process. The resulting meshes are crack-free, and all the faces are quadrilaterals. Limit surface of the resulting meshes is the same as the original limit surface. Test results show that the number of faces generated in the adaptively refined meshes is one order less than the uniform approach. The proposed technique works for cubic Doo-Sabin subdivision surfaces, non-uniform cubic subdivision s...

Estimating subdivision depth of Catmull-Clark surfaces

In this paper, both general and exponential bounds of the distance between a uniform Catmull-Clark surface and its control polyhedron are derived. The exponential bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Based on the exponential bound, we can predict the depth of subdivision within a user-specified error tolerance. This is quite useful and important for pre-computing the subdivision depth of subdivision surfaces in many engineering applications such as surface/surface intersection, mesh generation, numerical control machining and surface rendering.

Estimating subdivision depths for rational curves and surfaces

An algorithm to estimate subdivision depths for rational curves and surfaces is presented. The subdivision depth is not estimated for the given curve/surface directly. The algorithm computes a subdivision depth for the polynomial curve/surface of which the given rational curve/surface is the image under the standard perspective projection. This subdivision depth, however, guarantees the required flatness of the given curve/surface after the subdivision. This work has applications in surface rendering, surface/surface intersection, and mesh generation.

Recursive Patterns or the Garden of Forking Paths

Patterns are presented that are reflections of well-known computer structures, and the procedures used to obtain these patterns are described The basic program, organized around a stack, is shown to be simple, allowing many patterns to be generated, only a few of which have been given here. A casual glance at the patterns illustrating this article, and a thought devoted to its title, will be enough to give readers with some knowledge of computing techniques a fair idea of the principles on which the production of the patterns was based. These principles are, in fact, so elementary that it is very surprising such patterns have not previously been published. I must therefore at the outset offer an apology in case I have overlooked earlier papers during my literature search. I did, however, find one reference to the basic procedure in The Sense of Order (1). There Gombrich offers the beginning of a simple recursive pattern as an illustration of the 'filling-in' procedure in decorative art. He then elaborates on the incentive-the 'horror vacui'-that spurs the decorator to fill in any void in his design. Speaking for myself, I must admit that I could not detect any personal 'fear of the void' while I was planning the computer programs for producing the patterns, nor when I was watching them being drawn by the machine. Rather, if at those moments I experienced any urge to fill in the voids in the plotter paper or on the graphics screen, this feeling was closer to the 'amor infiniti' suggested by Gombrich, than to any fear. The recursive patterns I am presenting here also display a tendency to fill the space in which they develop. The computer program achieves this effect by closing the gaps between lines as the density of lines grows. However, between certain lines the gaps always remain wider than elsewhere. These particular gaps gain significance with increasing density: at lower levels of density the attention is drawn to the lines rather than the gaps, but as density increases, attention is diverted to the gaps in the pattern, which becomes increasingly fractured. Finally, the pattern of lines dissolves itself into a texture, broken only by the pattern, the significant parts of which are now the voids. Computers, automatic drafting machines and graphic screens were the only tools used to generate these patterns. In one sense, they were 'created' by a computer program and, subject to previous claims, 'discovered' by the author. As I do not think that they should be classified as 'Computer Art', I asked the editors of Leonardo to publish this article as 'Computer Graphics'. Most of the specimens that could appropriately be classified as Computer Art are currently produced in either of two modes: by an artist conversing with a computer via a graphics screen, the computer being in this case the tool used in a deliberate composition; and random processes, introduced as a kind of substitute for an artist's intuition, producing a variety of specimens, which the artist later evaluates aesthetically. In my case the program is operated in 'batch' mode, which is not conversational, and no randomness is involved. The program reads a few parameters, such as the depth of subdivision, the choice of colours, etc., and then produces the patterns without