Geometry of the ringed surfaces in R4R4 that generate spatial Pythagorean hodographs

A Pythagorean-hodograph (PH) curve  r(t)=(x(t),y(t),z(t))r(t)=(x(t),y(t),z(t)) has the distinctive property that the components of its derivative r′(t)r′(t) satisfy x′2(t)+y′2(t)+z′2(t)=σ2(t) for some polynomial σ(t)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)x′(t),y′(t),z′(t) in terms of polynomials u(t),v(t),p(t),q(t)u(t),v(t),p(t),q(t) through a construct that can be interpreted as a mapping from R4R4 to R3R3 defined by a quaternion product or the Hopf map. Under this map, r′(t)r′(t) is the image of a ringed surface S(t,ϕ)S(t,ϕ) in R4R4, whose geometrical properties are investigated herein. The generation of S(t,ϕ)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)S(t,ϕ) are derived. Furthermore, if r′(t)r′(t) is non-degenerate, S(t,ϕ)S(t,ϕ) is not developable (a non-trivial fact in R4R4). 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)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.