Generalized Expo - Rational B - Splines

The concept of expo-rational B-spline (ERBS) was introduced in [6] where the ’superproperties’ of ERBS were studied for the first time. In 2006 the first author of the present paper proposed a framework for the generalization of ERBS, the so-called generalized ERBS (GERBS), and considered one important instance of GERBS, the so-called Euler Beta-function B-splines (BFBS) which offered a good trade-off between preservation of the important ’superproperties’ of ERBS (with some reductions) and easy computability (BFBS being explicitly and exactly computable piecewise polynomials, while ERBS were computed by a very rapidly converging, yet approximate, numerical integration algorithm). The practical performance of BFBS was tested against classical polynomial Bsplines and ERBS in some MSc Diploma Thesis projects at the R&D Group for Mathematical Modelling, Numerical Simulation and Computer Visualization at Narvik University College, resulting in the development of a related software application in [19]. The first detailed exposition of the concepts and theory related to GERBS and BFBS was in [5] at the Seventh International Conference on Mathematical Methods for Curves and Surfaces in Tønsberg, Norway, in 2008. Received: July 8, 2009 c © 2009 Academic Publications Correspondence author 834 L.T. Dechevsky, B. Bang, A. Laks̊a The purpose of the present paper is to provide a detailed systematic exposition of the definitions, basic properties and advanced features and ’superproperties’ of GERBS and BFBS, to trace the evolution of these properties as the consideration is being gradually focused from the most general concept of GERBS as reparametrizations of the piecewise-affine B-splines, with bounded Jordan variation, through the justification of the introduction of BFBS as Csmooth GERBS, m ∈ N, to the ultimate construction of ERBS as C-smooth GERBS. In the course of the exposition we keep track of the trade-off between the computability of the GERBS versus the extension of the range of its ’superproperties’. At the same time, we compare the features of GERBS with those of classical polynomial Schoenberg B-splines. The main new mathematical concepts in the study are being geometrically elucidated via visualization plots generated by our own in-house software applications. Several new special relevant topics are included in the study, as follows. • comparison between the computational complexity of interpolation, approximation and numerical linear algebraic problems when solved by classical polynomial B-splines and by (G)ERBS, in sections 2.1.3 and 2.2.1; • heuristic motivation for introducing GERBS, complementary to the heuristics for introducing ERBS [6], in section 4; • several new model constructions of GERBS, including, but not limited to, BFBS, in section 5; • a concluding discussion about the considerable advantages of replacing NURBS (Non Uniform Rational B-splines) by rational forms of GERBS as universal tools of Isogeometric Analysis, in section 6. This is the first in a sequence of papers dedicated to the design and properties of GERBS, as functions of one and several variables, in one and several dimensions. AMS Subject Classification: 41A15, 65D07, 33B15, 33B20, 33F05, 41A30, 65D05, 65D10, 65D20, 65D30