G¹ Hermite interpolation method for spatial PH curves with PH planar projections

The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G¹ Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G¹ interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G¹ data in general, however, we show the feasibility of the system for the G¹ data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.