$C^2$ Hermite interpolation by Pythagorean Hodograph space curves

We solve the problem of $C^2$ Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of $C^2$ space boundary data (two points with associated first and second derivatives) we construct a four-dimensional family of PH interpolants of degree $9$ and introduce a geometrically invariant parameterization of this family. This parameterization is used to identify a particular solution, which has the following properties. First, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Second, it has the best possible approximation order 6. Third, it is symmetric in the sense that the interpolant of the ``reversed'' set of boundary data is simply the ``reversed'' original interpolant. This particular PH interpolant is exploited for designing algorithms for converting (possibly piecewise) analytical curves into a piecewise PH curve of degree $9$ which is globally $C^2$, and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve. The algorithms are presented along with an analysis of their error and approximation order.