Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Nonlinear pyramid transforms based on median-interpolation
SIAM Journal on Mathematical Analysis
Subdivision Methods for Geometric Design: A Constructive Approach
Subdivision Methods for Geometric Design: A Constructive Approach
Gliding spline motions and applications
Computer Aided Geometric Design
Continuous M-Estimators and Their Interpolation by Polynomials
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
Meshless geometric subdivision
Graphical Models
Subdivision Curves on Surfaces and Applications
CIARP '08 Proceedings of the 13th Iberoamerican congress on Pattern Recognition: Progress in Pattern Recognition, Image Analysis and Applications
Numerical Checking of C1 for Arbitrary Degree Quadrilateral Subdivision Schemes
Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII
A variational approach to spline curves on surfaces
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Computer Aided Geometric Design
On parametric smoothness of generalised B-spline curves
Computer Aided Geometric Design
Variations on the four-point subdivision scheme
Computer Aided Geometric Design
Approximation order of interpolatory nonlinear subdivision schemes
Journal of Computational and Applied Mathematics
Blending based corner cutting subdivision scheme for nets of curves
Computer Aided Geometric Design
Approximation order from stability for nonlinear subdivision schemes
Journal of Approximation Theory
Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces
Advances in Computational Mathematics
Analysis of subdivision schemes for nets of functions by proximity and controllability
Journal of Computational and Applied Mathematics
Journal of Approximation Theory
Geometric conditions for tangent continuity of interpolatory planar subdivision curves
Computer Aided Geometric Design
Beyond Catmull–Clark? A Survey of Advances in Subdivision Surface Methods
Computer Graphics Forum
Full length article: Invariance property of proximity conditions in nonlinear subdivision
Journal of Approximation Theory
Geodesic Polar Coordinates on Polygonal Meshes
Computer Graphics Forum
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Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the Riemannian sense or in a certain Lie group sense), or by projection of the affine averages onto a surface. The analysis of these schemes is based on their proximity to the linear schemes which they are derived from. We verify that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition. We further show that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property, and if the distances of consecutive points of the initial control polygon are small enough. Moreover, if S satisfies a smoothness condition which is sufficient for its limit curves to be C^1, and if T is convergent, then the curves generated by T are also C^1. Similar analysis of C^2 smoothness is postponed to a forthcoming paper.