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(t)+y(t)+z(t) = σ(t) for some polynomial σ(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 to R defined by a quaternion product or the Hopf map. Under this map, r(t) is the image of a ringed surface S(t, φ) in R, whose geometrical properties are investigated herein. The generation of S(t, φ) 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, φ) are derived. Furthermore, if r(t) is non–degenerate, S(t, φ) is not developable (a non–trivial fact in R). 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, φ). 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.