  In the paper, we show that the transformations between modified Jacobi and
Bernstein bases of the constrained space of polynomials of degree at most $n$
can be performed with the complexity $O(n^2)$. As a result, the algorithm of
degree reduction of B\'ezier curves that was first presented in (Bhrawy et al.,
J. Comput. Appl. Math. 302 (2016), 369--384), and then corrected in (Lu and
Xiang, J. Comput. Appl. Math. 315 (2017), 65--69), can be significantly
improved, since the necessary transformations are done in those papers with the
complexity $O(n^3)$. The comparison of running times shows that our
transformations are also faster in practice.
