Helical polynomial curves and double Pythagorean hodographs I. Quaternion and Hopf map representations

Abstract

For regular polynomial curves r ( t ) in R 3 , relations between the helicity condition, existence of rational Frenet frames, and a certain “double” Pythagorean-hodograph (PH) structure are elucidated in terms of the quaternion and Hopf map representations of spatial PH curves. After reviewing the definitions and properties of these representations, and conversions between them, linear and planar PH curves are identified as degenerate spatial PH curves by certain linear dependencies among the coefficients. Linear and planar curves are trivially helical, and all proper helical polynomial curves are PH curves. All spatial PH cubics are helical, but not all PH quintics. The two possible types of helical PH quintic (monotone and general) are identified as subsets of the PH quintics by constraints on their quaternion coefficients. The existence of a rational Frenet frame and curvature on polynomial space curves is equivalent to a certain “double” PH form, first identified by Beltran and Monterde, in which | r ′ ( t ) | and | r ′ ( t ) × r ′ ′ ( t ) | are both polynomials in t . All helical PH curves are double PH curves, which encompass all PH cubics and all helical PH quintics, although non-helical double PH curves of higher order exist. The “double” PH condition is thoroughly analyzed in terms of the quaternion and Hopf map forms, and their connections. A companion paper presents a complete characterization of all helical and non-helical double PH curves up to degree 7.