The elastic bending energy of pythagorean-hodograph curves Original Research Article

Pythagorean-hodograph (PH) curves admit closed-form expressions for the integral of the square of the curvature with respect to arc length (the “energy” integral) involving only rational functions, arctangents, and natural logarithms. In particular, the complex formulation of PH curves greatly facilitates the derivation of these expressions, yielding compact and efficient implementations in any high-level language that provides complex arithmetic. Explicit formulae are presented for the case of Tschirnhausenʹs cubic and the regular PH quintics, and in the latter case the use of the energy integral in optimizing the “fairness” of geometric Hermite interpolants is discussed. Compelling empirical evidence indicates that, for “reasonable” derivative data, first-order PH quintic Hermite interpolants are systematically of lower energy than their ordinary cubic counterparts.