Existence of Pythagorean-hodograph quintic interpolants to spatial G 1 Hermite data with prescribed arc lengths
Published Web Locationhttps://doi.org/10.1016/j.jsc.2019.02.008
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.