Curves and surfaces for computer aided geometric design: a practical guide
Curves and surfaces for computer aided geometric design: a practical guide
Rational continuity: parametric, geometric, and Frenet frame continuity of rational curves
ACM Transactions on Graphics (TOG) - Special issue on computer-aided design
Bezier and B-Spline Techniques
Bezier and B-Spline Techniques
Non-linear subdivision using local spherical coordinates
Computer Aided Geometric Design
A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics
Computer Aided Geometric Design
Exact Evaluation of Non-Polynomial Subdivision Schemes at Rational Parameter Values
PG '07 Proceedings of the 15th Pacific Conference on Computer Graphics and Applications
Nonlinear subdivision through nonlinear averaging
Computer Aided Geometric Design
Journal of Computational and Applied Mathematics
Subdivision Surfaces
Four-point curve subdivision based on iterated chordal and centripetal parameterizations
Computer Aided Geometric Design
A circle-preserving C2 Hermite interpolatory subdivision scheme with tension control
Computer Aided Geometric Design
Incenter subdivision scheme for curve interpolation
Computer Aided Geometric Design
Variations on the four-point subdivision scheme
Computer Aided Geometric Design
Approximation order from stability for nonlinear subdivision schemes
Journal of Approximation Theory
Graphical Models
A subdivision scheme for surfaces of revolution
Computer Aided Geometric Design
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The promise of modeling by subdivision is to have simple rules that avoid cumbersome stitching-together of pieces. However, already in one variable, exactly reproducing a variety of basic shapes, such as conics and spirals, leads to non-stationary rules that are no longer as simple; and combining these pieces within the same curve by one set of rules is challenging. Moreover, basis functions, that allow reading off smoothness and computing curvature, are typically not available. Mimicking subdivision of splines with non-uniform knots allows us to combine the basic shapes. And to analyze non-uniform subdivision in general, the literature proposes interpolating the sequence of subdivision control points by circles. This defines a notion of discrete curvature for interpolatory subdivision. However, we show that this discrete curvature generically yields misleading information for non-interpolatory subdivision and typically does not converge, not even for non-uniform spline subdivision. Analyzing the causes yields three general approaches for solving or at least mitigating the problem: equalizing parameterizations, sampling subsequences and a new skip-interpolating subdivision approach.