Geometry of the ringed surfaces in R4 that generate spatial Pythagorean hodographs. (March 2016) - Farouki, Rida T.
Abstract: 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 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, ϕ ) in R 4, 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 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, 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.