$\boldsymbol{G^2[C^1]}$ Hermite interpolation using septic PH curves

In this paper, we examine the problem of $G^2[C^1]$ Hermite interpolation using septic Pythagorean hodograph (PH) curves. PH curves are a special class of polynomial parametric curves, which have a polynomial arc length function and rational offsets, and are thus widely used in computer-aided design. According to different factorizations of their first derivative in complex form, septic PH curves are classified into three classes. The curves in the first class are all regular, and their construction under any $G^2[C^1]$ condition has already been studied. Therefore, in this paper we focus on the remaining two classes. The number of septic PH curves in the second class is even and no more than six. The existence of septic PH curves in the third class is dependent on the initial Hermite data, and users may specify a real parameter to determine the resultant curve. In addition, we provide the approximation of arcs with septic PH curves as examples demonstrating the application of our results.