In isogeometric analysis, it is frequently required to handle the geometric models enclosed by four-sided or non-four-sided boundary patches, such as trimmed surfaces. In this paper, we develop a Gregory solid based method to parameterize those models. First, we extend the Gregory patch representation to the trivariate Gregory solid representation. Second, the trivariate Gregory solid representation is employed to interpolate the boundary patches of a geometric model, thus generating the polyhedral volume parametrization. To improve the regularity of the polyhedral volume parametrization, we formulate the construction of the trivariate Gregory solid as a sparse optimization problem, where the optimization objective function is a linear combination of some terms, including a sparse term aiming to reduce the negative Jacobian area of the Gregory solid. Then, the alternating direction method of multipliers (ADMM) is used to solve the sparse optimization problem. Lots of experimental examples illustrated in this paper demonstrate the effectiveness and efficiency of the developed method.
A surface interpolation method for irregular meshes of curves is proposed. When a face in the mesh is interpolated, a surface patch on the face is individually generated from localized boundary information. The tangent planes of generated patches are continuous. This method employs an extended Bezier patch as a surface equation. The patch can be defined, specifying independently normal derivatives along the boundary curves. The total procedure of generating patches is simple and quick. Publisher Summary Smooth or blending function interpolants, which match a given function and slopes on the boundary of a rectangle or a triangle, usually require that the cross derivative or twist terms be defined unambiguously at vertices. Smooth interpolation schemes that avoid such restrictions could be useful for the piecewise generation of surfaces in computer-aided geometric design. This chapter discusses two such schemes—one over a rectangle and the other over a triangle. This chapter describes a new interpolation scheme for the triangle. This scheme has a relatively simple construction, it is symmetric in that each side of the triangle is treated in the same way, and it involves no compatibility constraints. The set of polynomials that are reproduced by an interpolation scheme is defined as the precision set of the interpolant. We propose a unified method of generating a wide range of three dimensional objects from polyhedra to solids with free-form surfaces. Modeling systems for polyhedra and systems for free-form surfaces have been developed independently in the past because of the difference in their underlying theory and practices. However, this is not desirable for a designer. So in this paper, we have shown a method in which a wide range of shapes are generated in one system by using local modifications. Local modifications are procedures used to change the shape of solids locally. The construction and the modification of three dimensional shapes by these procedures are natural and easy for a designer in many cases. The implementation of these procedures in a computer is easy and their execution does not require much time. Our method to construct a solid with free-form surfaces consists of following three phases. 1) A solid which serves as a basis of free-form shape design is generated by local modifications. Edges of this solid are straight lines but its faces are not necessarily flat planes. 2) From this model, a curve model which adequately represents the characteristics of a free-form shape is generated. 3) Surface equations interpolating over the curve model are generated. We have made a geometric modeling system MODIF. Using this system, a complicated solid with free-form surfaces can be designed easily. MODIF can generate color shaded pictures and cutter path data for making a real object model by NC machining tool. Most existing meshing algorithms for a 2D or shell figure requires the figure to have exactly four sides. Generating structured grids in the n-sided parametric region of a trimmed surface thus usually requires to first partition the region into four-sided sub-regions. We address the automatic structured grid generation problem in an n-sided region by fitting a planar Gregory patch so that the partition requirement is naturally avoided. However, self-overlapping may occur in some portions of the algebraically generated grid; this severely limits its usage in most of engineering and scientific applications where a grid system with no self-intersecting is strictly required. To solve the problem, we use a functional optimization approach to move grid nodes in the u−v domain of the trimmed surface to eliminate the self-overlapping. The derivatives of a Gregory patch, which are extremely difficult to compute analytically, are not required in our method. Thus, our optimization algorithm compares favourably at least in terms of speed with some other mesh optimization algorithms, such as the elliptic PDE method. In addition, to overcome the difficulty of guessing a good initial position of every grid node for the conjugate gradient method, a progressive optimization algorithm is incorporated in our optimization. Experiment results are given to illustrate the usefulness and effectiveness of the presented method. Copyright © 2004 John Wiley & Sons, Ltd. We describe the theoretical frame for a method of creating and describing rounded objects of arbitrary topology in CAD, and its implementation for UNIGRAFIX, a polygon-based modeler developed at UC Berkeley that generates black-and-white, smooth-shaded images on several output devices. The mathematical foundation for building triangular patches interpolating cubic edges and blending with geometric continuity is given, and various approaches are discussed. To represent curvature information, we extended the UNIGRAFIX language to UniCubix, and we implemented uci, an interactive shell that interprets a UniCubix description and converts it into UNIGRAFIX wireframes or polyhedral nets that approximate curved patches. Uci also provides a prototype of a global smoothing operation, that takes a polyhedral object of arbitrary topology and creates the UniCubix representation of a smooth object interpolating the input vertices.