On the derivative formula of a rational Bézier curve at a corner

Keywords: Arbitrary order derivative Derivative formula Rational Bézier curve a b s t r a c t In this paper, we develop the exact arbitrary order derivative formula of a rational Bézier curve at its corner. Some examples are presented to demonstrate the computation of the exact derivatives. Derivative formula of a parametric curve has important applications in geometric design and applied mathematics. For example, to stitch two Bézier curves smoothly, we need the derivative formulae at their merged corners as constraints to configure the control points at either side of the corner. Although computing the derivatives of a polynomial Bézier curve is straightforward, the derivative formula of a rational Bézier curve is very complicated and difficult to represent. To demonstrate its complexity, we develop the third order derivative at the corner t ¼ 0 of the rational curve (1) (refer to Section 2), that is,