Characterization of line width variation
Alfred K. Wong, Antoinette F. Molless, et al.
SPIE Advanced Lithography 2000
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.
Alfred K. Wong, Antoinette F. Molless, et al.
SPIE Advanced Lithography 2000
Peter Wendt
Electronic Imaging: Advanced Devices and Systems 1990
Salvatore Certo, Anh Pham, et al.
Quantum Machine Intelligence
A.R. Conn, Nick Gould, et al.
Mathematics of Computation