Euclidean and Minkowski Pythagorean hodograph curves over planar cubics

Starting with a given planar cubic curve [x(t), y(t)]T, we construct Pythagorean hodograph (PH) space curves of the form [x(t), y(t), z(t)]T in Euclidean and in Minkowski space, which interpolate the tangent vector at a given point. We prove the existence of these curves for any regular planar cubic and we express all solutions explicitly. It is shown that the constructed curves provide upper and lower polynomial bounds on the parametrical speed and the arc–length function of the given cubic. We analyze the approximation order and derive an explicit formula for the gap between the bounds. In addition, we discuss the approximation of the offset curves. Finally we define an invariant which measures the deviation of a given planar cubic from being a PH curve.