Curvatures of the quadratic rational Bézier curves

We find necessary and sufficient conditions for the curvature of a quadratic rational Bezier curve to be monotone in [0, 1], to have a unique local minimum, to have a unique local maximum, and to have both extrema in (0, 1), and we also visualize them in figures. As an application, we present a necessary and sufficient condition for the offset curve to be regular and to have the same tangent direction with the given quadratic rational Bezier curve, and give a simple algorithm to find it.