Derivation of the Beta-spline Curve Representation

Like a B-spline [2, 20], a Beta-spline curve or surface is specified by a set of points called control vertices. Although these vertices do not generally lie on the generated curve or surface, their positions completely determine its shape. The vertices for a curve are an ordered sequence and are connected in succession to form a (closed or open) control polygon (Figure 7.1). This sequence of vertices will be denoted $${\text{V = }}\left[ {{\text{V}}_0 ,{\text{V}}_1 , \ldots ,{\text{V}}_m } \right]$$ .