Parametric PDE techniques, which use partial differential equations (PDEs) defined over a 2D or 3D parametric domain to model graphical objects and processes, can unify geometric attributes and functional constraints of the models. PDEs can also model implicit shapes defined by level sets of scalar intensity fields. In this paper, we present an approach that integrates parametric and implicit t...

In the area of automatic parallelization of programs, analyzing and transforming loop nests with parametric aane loop bounds requires fundamental mathematical results. The most common geometrical model of iteration spaces, called the polytope model, is based on mathematics dealing with convex and discrete geometry, linear programming, combinatorics and geometry of numbers. In this paper, we pre...

  This paper demonstrates a preliminary investigation of geometry, function and its relation to DFX principles, namely DFM (Design for Manufacturing). This is the starting point for research on the development of an expert system that assesses design goals along DFX principles in a feature-based CAD environment. There is a need for a deeper level of understanding of the relationship between geo...

This paper summarizes the results of investigations into the development of parametric waverider geometry models, with emphasis on their efficiency, in terms of their ability to cover a large feasible design space with a sufficiently small number of design variables to avoid the ‘curse of dimensionality’. The work presented here is focused on the parameterization of idealized waverider forebody...

We study the geometry of biopolymer networks and effects of the geometry on bulk mechanical properties. It is shown numerically that the physical network geometry can be quantified statistically and regenerated from its statistical description, so that the regenerated network exhibits the same network mechanics as the physical network in the elastic regime. A collagen-I biopolymer network is us...

Normal vectors of a surface are essential in various geometry-related applications. Since a normal vector at a point on a parametric surface is computed by a cross product of two tangent vectors at the point, a regular parametric surface may possess a degenerate normal vector at a point when tangent vectors of the surface are linearly dependent at the point. This paper describes an algorithm wh...

We investigate existence and properties of discrete mixture representations $P_{\theta }=\sum _{i\in E}w_{\theta }(i)\,Q_{i}$ for a given family }$, $\theta \in \Theta $, probability measures. The noncentral chi-squared distributions provide classical example. obtain results about geometric statistical aspects the problem, latter including loss Fisher information, Rao-Blackwellization, asymptot...

In the literature of fuzzy measures, there exist many well known parametric and non-parametric measures, each with its own merits and limitations. But our main emphasis is on applications of these measures to a variety of disciplines. To extend the scope of applications of these fuzzy measures to geometry, we need some special fuzzy measures. In this communication, we have introduced two new fu...