Extending B-spline tools and algorithms to geometrically continuous splines: A study of similarities and differences

This paper continues the exploration of geometrically continuous splines begun in (Dyn and Micchelli, 1988; Barry et al., 1991; Barry et al., 1993). Here we consider the question "to what extent are the fundamental tools and algorithms derived for arbitrary degree geometrically continuous splines in (Seidel, 1993; Barry et al., 1993) similar to the tools and algorithms for B-spline curves, and to what extent are they different?" To explore this question we present new results in four areas - (i) explicit formulas for dual dunctionals for geometrically continuous B-splines, (ii) complexity of the combinations in the algorithms, (iii) recurrences induced by these algorithms, and (iv) progressive curves in the geometrically continuous setting. Each of these areas illustrates the similarities and differences between the tools and algorithms for geometrically continuous splines and tools and algorithms for B-spline curves.