IJPAM: Volume 65, No. 3 (2010)

GEOMETRIC MODELLING
WITH BETA-FUNCTION B-SPLINES, II:
TENSOR-PRODUCT PARAMETRIC SURFACES

Arne Lakså$^1$, Børre Bang$^2$, Lubomir T. Dechevsky$^3$
$^{1,2,3}$R&D Group for Mathematical Modelling,
Numerical Simulation and Computer Visualization
Faculty of Technology
Narvik University College
2, Lodve Lange's Str., P.O. Box 385, N-8505, Narvik, NORWAY
$^1$e-mail: ala@hin.no
url: https://ansatte.hin.no/ala/
$^2$e-mail: bb@hin.no
url: https://ansatte.hin.no/bb/
$^3$e-mail: ltd@hin.no
url: https://ansatte.hin.no/ltd/


Abstract.This is the second paper in a sequence of several articles dedicated to the development of applications of Euler Beta-function B-splines (BFBS) to Computer-aided Geometric Design (CAGD) and, in particular, for geometric modelling of parametric curves, surfaces and volume deformations. This study is an analogue of the study conducted in [#!Laksa:2005!#,#!AlPhD:2006!#] for the case of expo-rational B-splines (ERBS). An important objective of this study is the comparison between the graphical and computational performance of BFBS versus ERBS, as well as the comparison of BFBS versus classical polynomial Schoenberg B-splines.

In the first paper [#!lbd:2010_1!#] of the sequence we discussed parametric curve interpolation based on BFBS, as well as the respective BFBS-based Bezier-type form of the resulting parametric curves. In the present paper we shall continue this topic by studying interpolation of bivariate BFBS tensor-product parametric surfaces and trivariate BFBS tensor-product volume deformations in 3D and the respective BFBS-based Bezier-type representations of the parametric surfaces and volume deformations resulting from the interpolation.

Received: March 23, 2010

AMS Subject Classification: 53A05, 65D05, 65D07, 65D17, 26A63, 26B15, 28A75, 33B15, 33B20, 33F05, 41A15, 41A30, 42A38, 44A10, 50A30, 51M25, 52A38, 53A04, 53A15, 53A17, 53A20, 65D10, 65D20, 65D30

Key Words and Phrases: geometric modelling, parametrization, curve, surface, volume deformation, tensor product, computer-aided geometric design, spline, B-spline, polynomial, exponential, rational, expo-rational, generalized, Euler Beta-function, complete, incomplete, Gamma-function, scalar, vector, point, matrix, piecewise, linear, affine, barycentric, convex, continuous, differentiable, smooth, Hermite interpolation, monomial basis, Bernstein basis, Bezier curve, control polygon, de Casteljau algorithm, Cox-de Boor algorithm, iterative, local, global, constant, variable, functional

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 65
Issue: 3