Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures

The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed. Conditions on the data that identify when the “ordinary” cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the interpolants depends on only one of the parameters, and that four (general) helical PH quintic interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how “close” the PH quintic interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the “ordinary” cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.