Controlling Surfaces Using the Shape Parameters

Abstract

Analogous to the demonstration of Beta-spline curves in Chapter 17, this chapter illustrates the use of the shape parameters in controlling surfaces. The effect of bias is shown in Figures 18.1, 18.2, and 18.3. These figures show Beta-spline surfaces with uniform shape parameter values of β1 = 1 with β2 = 0, of β1 = 2 with β2 = 0, and of β1 = 0.5 with β2 = 0, respectively. A Beta-spline surface where both β1 and β2 are zero is illustrated in Figure 18.4. Increasing uniform tension values of β2 = 5, β2 = 10, β2 = 20, β2 = 50, β2 = 100, and β2 = 200 with a uniform bias value of β1 = 1 are shown in Figures 18.5 through 18.10. Negative tension values greater than −k are illustrated in Figures 18.11 through 18.16 where β2 = −4, β2 = −5, β2 = −6, β2 = −7, β2 = −8, and β2 = −9, respectively, with β1 = 1. Tension values which are more negative (that is, less than −k) are shown in Figures 18.17 through 18.20, where β2 = −25, β2 = −30, β2 = −50, and β2 = −100, respectively, with β1 = 1. Continuous shape parameters are illustrated in Figures 18.21 through 18.24. Here, β1 is uniformly unity and β2 is zero at all the joints except the one that corresponds to the top vertex. At the joint, β2 = 2, β2 = 5, β2 = 10, and β2 = 20 in Figures 18.21 through 18.24, respectively.