Planar class A Bézier curves: The case of real eigenvalues

We consider planar, special Bézier curves, i.e., polynomial Bézier curves in the plane whose control polygon is fully identified by the first edge and a 2×2 matrix M. We focus on the case where M has two real eigenvalues and we formulate, in terms of the Schur form of M, necessary and sufficient conditions for a regular, planar special Bézier curve to be a class A curve, i.e., a curve with monotone curvature, for any degree and any choice of the first edge. The result is simple in its formulation and can thus be easily used for both designing class A curves and analyzing given special Bézier curves.