**Branch :**Computer Science and Engineering

**Subject :**Computer graphics

**Unit :**THREE DIMENSIONAL GRAPHICS

## Beta-Spline

**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 P_{1-j }(u) and P_{J-1}(u) (Fig. 10-47). Zero-order continuity (positional continuity), G^{0}, at u, is obtained by requiring

First-order continuity (unit tangent continuity), G1, is obtained by requiring tangent vectors to be proportional:

**Β _{1}P’_{j-1}(u1) = P’_{j}(u_{j}), β_{1}> 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

**β _{1}P’’_{j-1}(uj) β_{2}P’’_{j-1}(uj) = P’’_{j}(uj)**

where**β _{2}** can be assigned any real number, and

**β**> 0. The curvature vector provides a measure of the amount of bending of the curve at position u,. When

_{1}**β**= 1 and

_{1}**β**= 0, beta-splines reduce to B-splines. Parameter

_{2}**β**is called the bins parameter since it controls the skewness of the curve. For

_{1}**β**> 1, the curve tends to flatten to the right in the direction of the unit tangent vector at the knots. For 0 <

_{1}**β**< 1, the curve tends to flatten to the left. The effect of

_{1}**β1**on the shape of the spline curve is shown in Fig. 10-48.

Parameter**β _{2}**is called the tension parameter since it controls how tightly or loosely the spline fits the control graph. As

**β2**increases, 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 = **β _{2}** 2

**β**4

_{1}^{3}**β**

_{1}^{2}4

**β**2.

_{1}
We obtain the B-spline matrix M_{B} when **β _{1}** = 1 and

**β**= 0. And we get the

_{2}**β**-spline with tension matrix M

_{Bt},when