International Journal of Shape Modeling

This is mainly a review article on magnetization dynamics in superfluid helium-3 within the framework of the Leggett–Takagi model. The mathematical setting of the latter is formulated. The problem of resonance relaxation, important for the physics of superfluid helium-3, is discussed from the point of view of the KAM theory. Numerical modeling is employed for studying the Arnold diffusion which, as is shown, could be the cause of catastrophic relaxation, observed experimentally. We suggest that similar methods may be employed for studying topology of invariant KAM tori.

Cyclic and Hochschild homology HH_*(X), HC_*(X) of a topological space X has appeared to be of wide interest since the publication of the papers by Burghelea [5], Burghelea and Fiedorowicz [6], and Goodwillie [12]. As far as HH_*(X) can be identified to and HC_*(X) to (the homology of a free loop space and the homology of the associated bundle [12]), cyclic and Hochschild homology provide the powerful technique of studying a free loop space. The computation of Hochschild homology is closely connected with the investigation of closed geodesics on manifolds and related dynamical systems. The computation of cyclic homology is connected with the groups of diffeomorphism of manifolds. It should be mentioned that the investigation of various topological invariants of is very important in view of their role in mathematical physics [27].

This paper presents a technique for designing a fillet joining two objects along two rail contours, one on each object's surface. Most fillet generation methods join surfaces which are either intersecting or in close proximity. They are ill-suited for joining farther-apart objects smoothly. Our method takes into account the relative spatial orientations and the neighborhoods of the rail contours so as to achieve a smooth join. We present a general reparametrization procedure for establishing correspondence between rail contours of arbitrary orientations. Intermediate contours defining the cross sectional shape of the fillet are designed through specifying a shape control parameter and two scaling functions. Finally, a lofting surface connecting the contours, and is at tangent continuity with the base surfaces, is generated. Objects of complex topological shapes can be created by joining handles and branches as fillets, thus ensuring topological integrity.

Since the body shapes of industrial products contain various kinds of free-form surfaces, designers need a 3D CAD system with which they can easily design such surfaces. This paper proposes a new method of modeling the free-form surface with a NURBS boundary Gregory patch. This patch, bounded by four NURBS curves, can be easily connected to other surfaces with G¹ continuity. In this paper, two methods that modify the surface shapes are described. One method modifies the direction and length of a tangent vector on the surface boundary, and the other controls the center area shape of the surface by moving a point on the surface. We have developed these capabilities on the solid modeling toolkit called DESIGNBASE. This system enables designers to dynamically control the surface shape, as they observe the parametric lines and curvature radii.

The construction of high-fidelity digital terrain models (DTM) requires the development of adaptive techniques for modeling and analysis of terrain surfaces. The paper presents a conceptual framework for digital terrain modeling which includes all relevant tasks to achieve this goal. This framework consists of fundamental techniques for the generation, manipulation, interpretation, and visualization of DTMs. Whereas generation and manipulation are modeling tasks, interpretation and visualization serve as analysis tools. Since modeling and analysis cannot be treated independently, adaptive modeling requires the integration of techniques from both areas. The role and relationship between modeling and analysis is discussed via examples dealing with sampling, triangulation, interpolation, filtering, joining and merging of DTMs, and feature extraction. Of the various tasks in digital terrain modeling two techniques are described in detail: adaptive interpolation from contour lines, and the extraction of channels and ridges from gridded DTMs. The functions developed within these two topics follow rather pragmatic approaches, including knowledge about the morphology of terrain surfaces, and clearly show the benefit of using adaptive methods.

In geometric modeling, numerous combinatorial structures have been proposed for modeling the topology of subdivisions. Often, these structures are deduced from analyses of the topology of cellular subdivisions of E³ when main combinatorial structures defined in combinatorial topology are simplicial. Here, we present a general framework for the definition of combinatorial structures, for the representation of the topology of triangulable cellular objects. Starting from semi-simplicial sets, which are well known structures in combinatorial topology, we define a simple mechanism, inspired by barycentric triangulation, leading to the definition of a wide class of combinatorial cellular complexes. Other combinatorial structures are deduced for particular subclasses of cellular complexes, using other mechanisms. Our approach is a purely combinatorial one. Advantages and drawbacks are discussed. For instance, relations between these combinatorial structures and geometric objects are deduced from that with semi-simplicial sets. Many important topological properties can be computed on these structures. For particular subclasses of cellular complexes, we can deduce equivalent structures, where less information is explicitly represented, decreasing thus the data storage complexity, and, sometimes, the time complexity of algorithms. One of the main drawbacks is the fact that cells of so-defined cellular complexes are always contractible into a vertex, due to the mechanism inspired from barycentric triangulation. Relations with other works are also discussed.