On the derivatives of rational B\'{e}zier curves

By studying the existing higher order derivation formulas of rational B\'{e}zier curves, we find that they fail when the order of the derivative exceeds the degree of the curves. In this paper, we present a new derivation formula for rational B\'{e}zier curves that overcomes this drawback and show that the $k$th degree derivative of a $n$th degree rational B\'{e}zier curve can be written in terms of a $(2^kn)$th degree rational B\'{e}zier curve.we also consider the properties of the endpoints and the bounds of the derivatives.