New developments in theory,algorithms and applications for Pythagorean-hodograph curves

The past decade has witnessed sustained interest in elucidating the basic theory of Pythagorean–hodograph (PH) curves, developing construction algorithms, formulating generalizations, and investigating applications. The rapid pace of this activity, encompassing diverse lines of inquiry, makes it desirable at this point to take a broad perspective of these recent developments, and assess their relationships. In the present article, we aim to address this need by categorizing recent results into a number of broad themes—extensions and specializations of the basic polynomial PH curves; rational orthonormal frames along spatial PH curves; construction and analysis algorithms for PH curves; surface design based on PH curves; and the use of PH curves in practical applications.