Detecting symmetries in polynomial Bézier curves

In a recent article, Alcázar (2014) presents algorithms for detecting central and mirror symmetries in planar polynomial curves, expressed with proper parameterization in the monomial (i.e., power) basis. However, for practical purposes in Computer Graphics and CAGD, the usual choice is the Bernstein–Bézier representation, because of its superior numerical and geometric characteristics. We point out that, in this form and for properly parameterized curves, detecting symmetry amounts to simply checking that the Bézier points exhibit pairwise symmetry. This result is a direct consequence of well-known properties of the Bézier representation, namely its symmetry, affine invariance, and uniqueness for proper parameterizations. Detecting the existence of a symmetric segment in a Bézier curve also amounts to a simple task, by analysing the last non-vanishing derivatives. Finally, these results carry over in a straightforward manner to symmetries in Euclidean space.