Nonlinear Weighted Average and Blossoming

Abstract

In this paper, we introduce a new averaging rule, the nonlinear weighted averaging rule. As an application, this averaging rule is used to replace the midpoint averaging in the de Casteljau evaluation algorithm and with this scheme we can also generate transcendental functions which cannot be generated by the classical de Casteljau algorithm. We also investigate the properties of the curves of the functions generated by blossoming, where the results show that these curves and the classical Bézier curves have some similar properties, including variation diminishing property and endpoint interpolation. However, the curves obtained by blossoming using nonlinear weighted averaging rules induced by certain functions violate some properties like convex hull property.

Keywords

Nonlinear weighted averaging rule / Midpoint averaging / De Casteljau algorithm / Bézier curves / Blossoming

References

[1] Aspert N, Ebrahimi T, Vandergheynst P. Non-linear subdivision using local spherical coordinates. Comput. Aided Geom. Des.. 2003, 20 EPFL–ARTICLE–86961 165-187

[2] Berger, M.: Géométrie 1. nathan, 1990. English edition: Geometry, vol. 1 (1987)

[3] Bezier P. The Mathematical Basis of the UNIURF CAD System. 2014 Oxford: Butterworth-Heinemann

[4] Cartan, H.: Cours de calcul différentiel (1977)

[5] Catmull E, Clark J. Recursively generated b-spline surfaces on arbitrary topological meshes. Comput. Aided Des.. 1978, 10 6 350-355

[6] Cohen A, Dyn N, Matei B. Quasilinear subdivision schemes with applications to eno interpolation. Appl. Comput. Harmon. Anal.. 2003, 15 2 89-116

[7] de Casteljau, P.: Splines focales. In: Curves and Surfaces in Geometric Design, pp. 91–103. Peters AK Wellesley (1994)

[8] De Casteljau PDF. Shape Mathematics and CAD. 1986 London: Kogan Page

[9] Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. In: Constructive Approximation, pp. 49–68. Springer (1989)

[10] Dyn N. Three families of nonlinear subdivision schemes. Stud. Comput. Math.. 2006, 12 23-38

[11] Farin GE, Farin G. Curves and Surfaces for CAGD: A Practical Guide. 2002 Burlington: Morgan Kaufmann

[12] Floater MS, Micchelli CA. Nonlinear Means in Geometric Modeling. 1997 Yorktown Heights: IBM Thomas J. Watson Research Division

[13] Gallier J, Gallier JH. Curves and Surfaces in Geometric Modeling: Theory and Algorithms. 2000 Burlington: Morgan Kaufmann

[14] Goldman R, Vouga E, Schaefer S. On the smoothness of real-valued functions generated by subdivision schemes using nonlinear binary averaging. Comput. Aided Geom. Des.. 2009, 26 2 231-242

[15] Goldman R. An Integrated Introduction to Computer Graphics and Geometric Modeling. 2009 Cambridge: CRC Press

[16] Lane JM, Riesenfeld RF. A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans. Pattern Anal. Mach. Intell.. 1980, 1 35-46

[17] Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics (1987)

[18] Marinov, M., Dyn, N., Levin, D.: Geometrically controlled 4-point interpolatory schemes. In: Advances in Multiresolution for Geometric Modelling, pp. 301–315. Springer (2005)

[19] Ramshaw L. Blossoming: A Connect-the-Dots Approach to Splines. 1987 Palo Alto: Digital Equipment Corporation

[20] Schaefer S, Vouga E, Goldman R. Nonlinear subdivision through nonlinear averaging. Comput. Aided Geom. Des.. 2008, 25 3 162-180

[21] Weyl H. The Classical Groups: Their Invariants and Representations. 2016 Princeton: Princeton University Press

Funding

CAS-TWAS