Employing the techniques presented by Nairn, Peters and Lutterkort in [1], sharp bounds are firstly derived for the distance between a planar parametric Bézier curve and a parameterization of its control polygon based on the Greville abscissae. Several of the norms appearing in these bounds are orientation dependent. We next present algorithms for finding the optimal orientation angle for which two of these norms become minimal. The use of these bounds and algorithms for constructing polygonal envelopes of planar polynomial curves, is illustrated for an open and a closed composite Bézier curve.