Learning Reduced Order Dynamics via Geometric Representations
Imran Nasim, Melanie Weber
SCML 2024
The Pythagorean hodograph (PH) curves are polynomial parametric curves {x(t), y(t)} whose hodograph (derivative) components satisfy the Pythagorean condition x'2(t)+y'2 (t) = σ22(t) for some polynomial σ(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. © 1995 American Mathematical Society.
Imran Nasim, Melanie Weber
SCML 2024
Andrew Skumanich
SPIE Optics Quebec 1993
M. Shub, B. Weiss
Ergodic Theory and Dynamical Systems
Sankar Basu
Journal of the Franklin Institute