Bulletin of the
Korean Mathematical Society BKMS

Representing planar Pythagorean hodograph (PH) curves by the complex roots of their hodographs, we standardize Farouki's double cubic method to become the undetermined junction point (UJP) method, and then prove the generic existence of solutions for general $C^1$ Hermite interpolation problems. We also extend the UJP method to solve $C^2$ Hermite interpolation problems with multiple PH cubics, and also prove the generic existence of solutions which consist of triple PH cubics with $C^1$ junction points. Further generalizing the UJP method, we go on to solve $C^2$ Hermite interpolation problems using two PH quintics with a $C^1$ junction point, and we also show the possibility of applying the modified UJP method to $G^2[C^1]$ Hermite interpolation.

Keywords: Pythagorean hodograph (PH) curve, complex representation, $C^1$($C^2$) Hermite interpolation, $G^2[C^1]$ Hermite interpolation, undetermined junction point (UJP) method

MSC numbers: Primary 68U05, 65D18, 68U07