A unified approach to subdivision algorithms near extraordinary vertices
Computer Aided Geometric Design
Interpolatory subdivision schemes and wavelets
Journal of Approximation Theory
Non-linear subdivision using local spherical coordinates
Computer Aided Geometric Design
Lofting curve networks using subdivision surfaces
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Convergence and C1 analysis of subdivision schemes on manifolds by proximity
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Subdivision schemes and attractors
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
SIAM Journal on Numerical Analysis
A 4-point interpolatory subdivision scheme for curve design
Computer Aided Geometric Design
A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computer Aided Geometric Design
Integration of CAD and boundary element analysis through subdivision methods
Computers and Industrial Engineering
Two open questions relating to subdivision
Computing - Geometric Modelling, Dagstuhl 2008
A generalized curve subdivision scheme of arbitrary order with a tension parameter
Computer Aided Geometric Design
Curvature of approximating curve subdivision schemes
Proceedings of the 7th international conference on Curves and Surfaces
A class of nonlinear four-point subdivision schemes
Advances in Computational Mathematics
Generalized Lane-Riesenfeld algorithms
Computer Aided Geometric Design
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We investigate a general class of nonlinear subdivision algorithms for functions of a real or complex variable built from linear subdivision algorithms by replacing binary linear averages such as the arithmetic mean by binary nonlinear averages such as the geometric mean. Using our method, we can easily create stationary subdivision schemes for Gaussian functions, spiral curves, and circles with uniform parametrizations. More generally, we show that stationary subdivision schemes for e^p^(^x^), cos(p(x)) and sin(p(x)) for any polynomial or piecewise polynomial p(x) can be generated using only addition, subtraction, multiplication, and square roots. The smoothness of our nonlinear subdivision schemes is inherited from the smoothness of the original linear subdivision schemes and the differentiability of the corresponding nonlinear averaging rules. While our results are quite general, our proofs are elementary, based mainly on the observation that generic nonlinear averaging rules on a pair of real or complex numbers can be constructed by conjugating linear averaging rules with locally invertible nonlinear maps. In a forthcoming paper we show that every continuous nonlinear averaging rule on a pair of real or complex numbers can be constructed by conjugating a linear averaging rule with an associated continuous locally invertible nonlinear map. Thus the averaging rules considered in this paper are actually the general case. As an application we show how to apply our nonlinear subdivision algorithms to intersect some common transcendental functions.