Definition of LR-Sequences for Trapezoidal Curves

Abstract

Consider the trapezoidal curve C_e(ζ) (including e=1) defined by Eqs. (1.1)–(.1.3) . With each point p = (a,b) ε C_e(ζ), we associate a point p⁽¹⁾ defined by p⁽¹⁾ = (b,c) ε C_e(ζ), and called the iterate of p [the coordinate c is uniquely determined by the requirement p⁽¹⁾ ε C_e(ζ)]. Each point p ε C_e(ζ) has a unique iterate point p⁽¹⁾, but several points may have the same iterate; for example, all points in the set {(a,eζ)|a ε I[e, 2–e]} have the same iterate point (eζ, 2ζ-eζ²). Also, (0,0)⁽¹⁾ = (2,0)⁽¹⁾ = (0,0). [Geometrically, the point p⁽¹⁾ is obtained from p by drawing a horizontal line from p to the point p′ on the 45°-line, followed by a vertical line from p′ to the point of intersection with C_e(ζ). This latter point is p⁽¹⁾.]