Hermite interpolation by Pythagorean hodograph quintics

Abstract

The Pythagorean hodograph (PH) curves are polynomial parametric curves { x ( t ) , y ( t ) } \{ x(t),y(t)\} whose hodograph (derivative) components satisfy the Pythagorean condition x ′ 2 ( t ) + y ′ 2 ( t ) ≡ σ 2 ( t ) x’{}^2(t) + y’{}^2(t) \equiv {\sigma ^2}(t) for some polynomial σ ( t ) \sigma (t) . Thus, unlike polynomial curves in general, PH curves have arc lengths and offset curves that admit exact rational representations. The lowest-order PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary first-order Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique result—there are always four distinct interpolants (of which only one, in general, has acceptable "shape" characteristics). We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the "good" interpolant (in terms of minimizing the absolute rotation number). This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used "ordinary" cubics.