© 2015 Elsevier Ltd. 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)=σ^{2}(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 R4 to R3 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 R4, 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 R4). 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.