Geometry of the ringed surfaces in R4 that generate spatial Pythagorean hodographs

A Pythagorean-hodograph (PH) curve r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) has the distinctive property that the components of its derivative r ' ( t ) satisfy x ' 2 ( t ) + y ' 2 ( t ) + z ' 2 ( t ) = ? 2 ( t ) for some polynomial ? ( t ) . Consequently, the PH curves admit many exact computations that otherwise require approximations. The Pythagorean structure is achieved by specifying x ' ( t ) , y ' ( t ) , z ' ( t ) in terms of polynomials u ( t ) , v ( t ) , p ( t ) , q ( t ) through a construct that can be interpreted as a mapping from R 4 to R 3 defined by a quaternion product or the Hopf map. Under this map, r ' ( t ) is the image of a ringed surface S ( t , ? ) in R 4 , whose geometrical properties are investigated herein. The generation of S ( t , ? ) through a family of four-dimensional rotations of a "base curve" is described, and the first fundamental form, Gaussian curvature, total area, and total curvature of S ( t , ? ) are derived. Furthermore, if r ' ( t ) is non-degenerate, S ( t , ? ) is not developable (a non-trivial fact in R 4 ). It is also shown that the pre-images of spatial PH curves equipped with a rotation-minimizing orthonormal frame (comprising the tangent and normal-plane vectors with no instantaneous rotation about the tangent) are geodesics on the surface S ( t , ? ) . Finally, a geometrical interpretation of the algebraic condition characterizing the simplest non-trivial instances of rational rotation-minimizing frames on polynomial space curves is derived.