A linear-time algorithm for computing a point on a polynomial or rational curve
in Bézier form with good geometric and numerical properties has been
recently given in [2]. This approach has also found applications in
accelerating the evaluation of Bézier surfaces and even B-spline curves (for
details, see [1]). We show that the method proposed in [2]
can be generalized to efficiently compute the quantities
$R_n'(t),R_n''(t),\ldots,R_n^{(k)}(t)$, where $R_n$ is a $d$-dimensional
rational Bézier curve of degree $n$ and $t\in [0,1]$. Moreover,
the algorithm may be adapted for a more general family of rational parametric
objects. Some remarks are given about applying it to Bézier surfaces.

Joint work with: Paweł Woźny (Institute of Computer Science, University of Wrocław, Poland)

[1] F. Chudy, New algorithms for Bernstein polynomials, their dual bases and B-spline functions, Ph.D.Thesis, University of Wrocław, 2022 (available on request).

[2] P. Woźny and F. Chudy, Linear-time geometric algorithm for evaluating Bézier curves, Computer Aided-Design 118 (2020), 102760.