On the curvature extrema of special cubic Bézier curves

with the special control point configuration P0 = Q0, P1 = (1 − a)Q0 + aQ1, P2 = aQ1 + (1− a)Q2, P3 = Q2 (2) has at most one local extremum of curvature when t ∈ (0, 1) and a ∈ ( 3 , 1]. 2 Special cases We need to treat two special cases first. When Q0 = Q2, the curve degenerates to a line segment, which has a kink (a point with infinite curvature) at t = 12 , so it clearly has exactly one local curvature extremum. With this handled, let us set, without loss of generality, Q0 = (−1, 0), Q1 = (b, h), Q2 = (1, 0), (3) where b, h ≥ 0. The second special case is when h = 0. Once again, the curve degenerates to a line segment. When b ∈ (−1, 1), its curvature is always 0, otherwise it has a single kink. In the following we will assume h > 0.