Computing surfaces invariant under subdivision

In this paper, we propose a general subdivision algorithm for generating surfaces. The algorithm has as motivation our earlier work on the design of free form curves where similar ideas were investigated. Here we describe some properties of uniform refinement algorithms for surface generation. A detail analysis of their properties will be given later by one of us.     

Education-driven research in CAD

We argue for a new research category, named education-driven research (EDR), which fills the gap between traditional field-specific research that is not concerned with educational objectives and research in education that focuses on fundamental teaching and learning principles and possibly on their customization to broad areas (such as mathematics or physics), but not to specific disciplines (such as CAD). The objective of EDR is to simplify the formulation of the underlying theoretical foundations and of specific tools and solutions in a specialized domain, so as to make them easy to understand and internalize. As such, EDR is a difficult and genuine research activity, which requires a deep understanding of the specific field and can rarely be carried out by generalists with primary expertise in broad education principles. We illustrate the concept of EDR with three examples in CAD: (1) the Split and Tweak subdivisions of a polygon and its use for generating curves, surfaces, and animations; (2) the construction of a topological partition of a plane induced by an arbitrary arrangement of edges; and (3) a romantic definition of the minimal and Hausdorff distances. These examples demonstrate the value of using analogies, of introducing evocative terminology, and of synthesizing the simplest fundamental building blocks. The intuitive understanding provided by EDR enables the students (and even the instructor) to better appreciate the limitations of a particular solution and to explore alternatives. In particular, in these examples, EDR has allowed the author to: (1) reduce the cost of evaluating a cubic B-spline curve; (2) develop a new subdivision curve that is better approximated by its control polygon than either a cubic B-spline or an interpolating 4-point subdivision curve; (3) discover how a circuit inclusion tree may be used for identifying the faces in an arrangement; and (4) rectify a common misconception about the computation of the Hausdorff error between triangle meshes. We invite the scientific community to encourage the development of EDR by publishing its results as genuine research contributions in peer-reviewed professional journals.     

Second-order accurate constraint formulation for subdivision finite element simulation of thin shells

We present a new method for enforcing boundary conditions within subdivision finite element simulations of thin shells. The proposed framework is demonstrated to be second-order accurate with respect to increasing refinement in the displacement and energy norm for simply supported, clamped, free and symmetric boundary conditions. Second-order accuracy on the boundary is consistent with the accuracy of subdivision-based approaches for the interior of a body. Our proposed framework is applicable to both triangular and quadrilateral refinement schemes, and does not impose any topological requirements upon the underlying subdivision control mesh. Several examples from an obstacle course of benchmark problems are used to demonstrate the convergence of the scheme. Copyright © 2004 John Wiley & Sons, Ltd.     

二阶A型Cooper变换的增强型

对二阶Ａ型Ｃｏｏｐｅｒ变换进行增强，放宽可用性条件，并证明了其正确性，最后举例说明其应用。     

Recursive calculation of ladder circuit frequency functions

It is shown how the transfer function as well as the input impedance and admittance of any linear time-invariant ladder circuit can be determined in closed form as frequency functions dependent upon the network element parameters. All elements of the ladder circuit must be explicitly specified by their frequency characteristics. The results are achieved with the aid of recursive algorithms. The outcome is demonstrated by means of an example.     

Pancyclicity of recursive circulant graphs

In this paper, we study the existence of cycles of all lengths in the recursive circulant graphs, and we show a necessary and sufficient condition for the graph being pancyclic and bipancyclic.     

A frame-based knowledge representation model and its mapping to nested relations

The first part of this paper introduces the basic constructs of a frame representation model and gives a formal definition for them. Subsequently the nested relational model (NF²) is described as an extension of the classical relational model to support relation-valued attributes. In the second part of this paper both models are compared with each other and a mapping of frame representation structures to NF² structures is specified. The structural similarities between frames and NF² relations are made clear and it is shown that their main difference is due to the type polymorphism introduced by the concept hierarchy of the frame model. This causes type collisions to occur in the strictly typed NF² model when frames are directly mapped to NF² structures. Two solutions to this problem are suggested and compared. The paper concludes with a reformulation of query operations of the frame model in terms of NF² algebra operations.     

Adaptive generation of surfaces in volume data

A widespread approach to generating polygonal approximations of iso-surfaces or contour surfaces in volume data is the socalled marching-cubes algorithm. This algorithm, however, has the disadvantage that the number of polygonal chains generated is considerable. The splitting-box algorithm presented here reduces the number of polygonal chains by adapting their size to the shape of the surface. The resulting polygonal chains offer a wide spectrum for representing the contour surface. An exact representation is achieved by a new type of generic patches calculated from the polygonal chains. Approximations of different quality may be obtained by combining the algorithm generating the patches with simple triangulations.     

An efficient hierarchicaltraversal algorithm for ray tracing

Ray tracing has been shown to be an excellent technique for rendering realistic images. However, it is important to reduce the lengthy computation time resulting from the brute-force nature of the original ray-tracing algorithms. In this paper, two ideas are proposed to speep up the well-known hierarchical subdivision method. First, a new hierarchy traversal scheme is described to reduce the number of raybounding volume intersection tests for secondary rays. Then, a plane-sweep method is proposed to make fewer intersection checks for eye rays. Experiments and discussions are presented to prove the feasibility of our methods.     

Remarks on the mean square tracking error

The purpose of this paper is to extend recent results of Ljung and Priouret to a more general class of regressors.