Existence of Pythagorean-hodograph quintic interpolants to spatial G~1 Hermite data with prescribed arc lengths

摘要

A unique feature of polynomial Pythagorean-hodograph (PH) curves is the ability to interpolate G(1) Hermite data (end points and tangents) with a specified total arc length. Since their construction involves the solution of a set of non-linear equations with coefficients dependent on the specified data, the existence of such interpolants in all instances is non-obvious. A comprehensive analysis of the existence of solutions in the case of spatial PH quintics with end derivatives of equal magnitude is presented, establishing that a two-parameter family of interpolants exists for any prescribed end points, end tangents, and total arc length. The two free parameters may be exploited to optimize a suitable shape measure of the interpolants, such as the elastic bending energy. (C) 2019 Elsevier Ltd. All rights reserved.