On the smooth convergence of subdivision and degree elevation for Bezier curves

Bezier subdivision and degree elevation algorithms generate piecewise linea r approximations of Bezier curves that converge to the original Bezier curv e. Discrete derivatives of arbitrary order can be associated with these pie cewise linear functions via divided differences. Here we establish the conv ergence of these discrete derivatives to the corresponding continuous deriv atives of the initial Bezier curve. Thus, we show that the control polygons generated by subdivision and degree elevation provide not only an approxim ation to a Bezier curve, but also approximations of its derivatives of arbi trary order. (C) 2001 Elsevier Science B.V. All rights reserved.