Geometry of the ringed surfaces in R4 that generate spatial Pythagorean hodographs. (March 2016)

• Record Type: Journal Article
• Authors: Farouki, Rida T.; Gutierrez, Robert
• Is Part Of: Journal of symbolic computation. Volume 73(2016)
• Page Start: 87
• Page End: 103
• Publication Date: 2016-03
• Subjects: Pythagorean-hodograph curves -- Quaternion polynomials -- Hopf map -- Four-dimensional geometry -- Ringed surfaces -- Gaussian curvature -- Rotation-minimizing frames; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.03.005
• Languages: English
• ISSNs: 0747-7171

Abstract

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.