Introduction: A generalization of Bsplines are the beta-splines, also referred to as Bsplines, that are formulated by imposing geometric continuity conditions on the first and second, parametric derivatives. The continuity parameters for beta-splines are called β parameters.
Beta-Spline Continuity Conditions: For a specified knot vector, we can designate the spline sections to the left and right of a particular knot uj with the position vectors P1-j (u) and PJ-1(u) (Fig. 10-47). Zero-order continuity (positional continuity), G0, at u, is obtained by requiring
First-order continuity (unit tangent continuity), G1, is obtained by requiring tangent vectors to be proportional:
Β1P’j-1(u1) = P’j(uj), β1> 1
Here, parametric first derivatives are proportional, and the unit tangent vectors are continuous across the knot.
Second-order continuity (curvature vector continuity), G2, is imposed with the condition
β1P’’j-1(uj) β2P’’j-1(uj) = P’’j(uj)
whereβ2 can be assigned any real number, and β1> 0. The curvature vector provides a measure of the amount of bending of the curve at position u,. When β1 = 1 and β2 = 0, beta-splines reduce to B-splines. Parameterβ1 is called the bins parameter since it controls the skewness of the curve. For β1> 1, the curve tends to flatten to the right in the direction of the unit tangent vector at the knots. For 0 <β1< 1, the curve tends to flatten to the left. The effect of β1on the shape of the spline curve is shown in Fig. 10-48.
Parameterβ2is called the tension parameter since it controls how tightly or loosely the spline fits the control graph. As β2increases, the curve approaches the shape of the control graph, as shown in Fig. 10-49.
Cubic, Periodic Beta-Spline Matrix Representation: Applying the beta-spline boundary conditions to a cubic polynomial with a uniform knot vector, we obtain the following matrix representation for a periodic beta-spline:
where S = β2 2β13 4β12 4β1 2.
We obtain the B-spline matrix MB when β1 = 1 and β2= 0. And we get the β-spline with tension matrix MBt,when