Simple regularity criteria for subdivision schemes
SIAM Journal on Mathematical Analysis
Composite primal/dual √3-subdivision schemes
Computer Aided Geometric Design
Toeplitz and circulant matrices: a review
Communications and Information Theory
Nonlinear subdivision through nonlinear averaging
Computer Aided Geometric Design
Polynomial reproduction by symmetric subdivision schemes
Journal of Approximation Theory
Computer Aided Geometric Design
NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes
ACM SIGGRAPH 2009 papers
Analysis and Design of Univariate Subdivision Schemes
Analysis and Design of Univariate Subdivision Schemes
Convergence and Smoothness of Nonlinear Lane–Riesenfeld Algorithms in the Functional Setting
Foundations of Computational Mathematics
A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
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The Lane-Riesenfeld algorithm for generating uniform B-splines provides a prototype for subdivision algorithms that use a refine and smooth factorization to gain arbitrarily high smoothness through efficient local rules. In this paper we generalize this algorithm by maintaining the key property that the same operator is used to define the refine and each smoothing stage. For the Lane-Riesenfeld algorithm this operator samples a linear polynomial, and therefore the algorithm preserves only linear polynomials in the functional setting, and straight lines in the geometric setting. We present two new families of schemes that extend this set of invariants: one which preserves cubic polynomials, and another which preserves circles. For both generalizations, as for the Lane-Riesenfeld algorithm, a greater number of smoothing stages gives smoother curves, and only local rules are required for an implementation.