Curvature continuous curves and surfaces
Computer Aided Geometric Design
Computer Aided Geometric Design
Making the Oslo algorithm more efficient
SIAM Journal on Numerical Analysis
An introduction to the use of splines in computer graphics
An introduction to the use of splines in computer graphics
Computer graphics and geometric modeling using Beta-splines
Computer graphics and geometric modeling using Beta-splines
Local Control of Bias and Tension in Beta-splines
ACM Transactions on Graphics (TOG)
Multiple-knot and rational cubic beta-splines
ACM Transactions on Graphics (TOG)
Knot insertion for Beta-spline curves and surfaces
ACM Transactions on Graphics (TOG)
ACM Transactions on Graphics (TOG)
Polar forms for geometrically continuous spline curves of arbitrary degree
ACM Transactions on Graphics (TOG)
Hierarchical B-spline refinement
SIGGRAPH '88 Proceedings of the 15th annual conference on Computer graphics and interactive techniques
IEEE Computer Graphics and Applications
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Goodman (1985) and Joe (1986) have given explicit formulas for (cubic) Beta-splines on uniform knot sequences with varying ß1 and ß2 values at the knots, and nonuniform knot sequences with varying ß2 values at the knots, respectively. The advantage of the latter formula is that it can also be used for knot sequences with multiple knots. Discrete Beta-splines arise when a Beta-spline curve is subdivided, i.e. the knot sequence is refined so that the curve is expressed in terms of a larger number of control vertices and Beta-splines. We prove that discrete Beta-splines satisfy the same properties as discrete B-splines, and present an algorithm for computing discrete Beta-splines and the new control vertices using the explicit formula of Joe (1986).