Planar projections of spatial Pythagorean-hodograph curves

Abstract

Although the orthogonal projection of a spatial Pythagorean–hodograph (PH) curve on to a plane is not (in general) a planar PH curve, it is possible to construct spatial PH curves so as to ensure that their orthogonal projections on to planes of a prescribed orientation are planar PH curves. The construction employs an analysis of the root structure of the components of the quaternion polynomials that generate spatial PH curves, and it encompasses both helical and non–helical spatial PH curves. An initial characterization for orthogonal projections of spatial PH curves on to the coordinate planes provides the basis for a generalization to projections of arbitrary direction, based on unit quaternion rotation transformations of R3.