We review the theory of homogeneous splines and their relationship to special rational splines considered by J. SS anchez{Reyes and independently by P. de Casteljau who called them focal splines. Applying an appropriate duality, we transform focal splines into a remarkable class of rational curves with rational oosets. We investigate geometric properties of these dual focal splines, and discuss...

We review the theory of homogeneous splines and their relationship to special rational splines considered by J. SS anchez{Reyes and independently by P. de Casteljau who called them focal splines. Applying an appropriate duality, we transform focal splines into a remarkable class of rational curves with rational oosets. We investigate geometric properties of these dual focal splines, and discuss...

Symbolic computation of NURBS plays an important role in many areas of NURBSbased geometric computation and design. However, any nontrivial symbolic computation, especially when rational B-splines are involved, would typically result in B-splines with high degrees. In this paper we develop degree reduction strategies for NURBS symbolic computation on curves. The specific topics we consider incl...

The matrix forms for curves and surfaces were largely promoted in CAD. These formulations are very compact to write, simple to program, and clear to understand. They manifest the desired basis as a matrix transformation of the common power basis. Furthermore, this implementation can be made extremely fast if appropriate matrix facilities are available in either hardware or software. Derivatives...

We explore extended B-splines as a stable basis for isogeometric analysis with trimmed parameter spaces. The stabilization is accomplished by an appropriate substitution of Bsplines that may lead to ill-conditioned system matrices. The construction for non-uniform knot vectors is presented. The properties of extended B-splines are examined in the context of interpolation, potential, and linear ...

This paper presents a novel framework for stochastic analysis of linear elastic fracture problems. Monte Carlo simulation (MCs) is adopted to address the multi-dimensional uncertainties, whose computation cost reduced by combination Proper Orthogonal Decomposition (POD) and Radial Basis Function (RBF). In order avoid re-meshing retain geometric exactness, isogeometric boundary element method (I...

Starting from the radiative transport equation we derive the scaling relationships that enable a single Monte Carlo (MC) simulation to predict the spatially- and temporally-resolved reflectance from homogeneous semi-infinite media with arbitrary scattering and absorption coefficients. This derivation shows that a rigorous application of this single Monte Carlo (sMC) approach requires the rescal...

Tchebycheffian splines are smooth piecewise functions whose pieces drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most the properties known in spline case. In particular, under suitable assumptions, admit representation terms basis functions, called B-splines (TB-splines), completely analogous to B-splines. A particularly ...

This work studied the lightweight design of an aircraft support based on lattice-filled structures. Different from traditional process structures, this combined several approaches, including topology optimization, homogenization analysis, and Non-Uniform Rational B-splines (NURBS) surface modeling, to reduce structural weight more effectively. The theories implementations involved in are introd...