5. A new direct method for the design of Bézier curves
Usually the designer of a Bézier curve sets up in an intuitive way a control polygon and inspects the resulting Bézier curve. If the curve is not as desired, then the control points are moved interactively one by one, until the result is satisfying. In the past there have also been invented a number of direct methods for curve design, that allow to change the whole Bézier curve by picking a point on the curve and changing its geometric constraints, e.g., its position in ambient space, the tangent direction and magnitude, or the magnitude of the curvature at the point (see for example Bartels and Beatty, 1989; Fowler and Bartels, 1993; Gleicher, 1992). The purpose of the present section is to indicate, how the additional freedom gained by the parameters a¯2,a¯3,... can be used to extend the known direct methods. Note that by the generalized de Casteljau algorithm of Theorem 4.6 these parameters do not describe local differential–geometric properties, but global properties of the generalized Bézier curves, e.g., an a¯2 > 0 and an a¯2 < 0 leads to a global decrease resp. increase (cf. Fig. 4) of curvature compared to the ordinary Bézier curves. We hope that other researchers can apply this approach to their particular problem. The new direct method is described and illustrated first from the perspective of the designer, then we discuss the mathematics behind the working steps. (Note, that also a weighting of control points can be done in the usual way in the generalized setup.)
