G1 Hermite interpolation method for spatial PH curves over planar PH curves
Yoonae Song  1@  , Soohyun Kim  2  , Hwan Pyo Moon  1  
1 : Dongguk University
2 : Applied Algebra and Optimization Research Center, Sungkyunkwan University

A special type of spatial Pythagorean hodograph (PH) curves, whose planar projection also has the PH
property, were introduced recently in [1]. In this work, we present a
G1 Hermite interpolation method for this
type of curves. While the projection plane is fixed as the
xy plane, these curves can be regarded as spatial PH
curves over planar PH curves, which we call PH over PH (PHoPH) curves. Because of the additional constraint,
PHoPH curves should have more complicated algebraic structure than usual spatial PH curves. We investigate
this structure using the quaternion algebra to obtain a compact representation of PHoPH from quaternion
generator polynomials. Based on this representation, we address the
G1 Hermite interpolation problem using
quintic PHoPH curves. The problem is formulated as a system of nonlinear equations involving trigonometric
functions, which can be solved by numerical methods. We analyze the feasibility of this problem and present
some computed examples.

[1] Farouki, Rida T and Knez, Marjeta and Vitrih, Vito and Zagar, Emil. Planar projections of spatial ˇ
Pythagorean-hodograph curves.
Computer Aided Geometric Design, 91:102049, 2021.

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