On the Bertrand Pairs of Open Non-Uniform Rational B-Spline Curves

Abstract

B-spline curves are used basically in Computer-Aided Design (CAD), Computer-Aided Geometric Design (CAGD), and Computer-Aided Modeling (CAM). In determining the invariants of curves and surfaces at any point, there are some difficulties in expressing it analytically and calculating its invariants at the desired point. For these curves and surfaces the way to overcome these difficulties is to design them with spline curves and surfaces. In this paper the second- and third-order derivatives of open Non-Uniform Rational B-Spline (NURBS) curves at the points t = t(d), t = t(m-d), and arbitrary point in domain of these curves are given. In addition, the Frenet vector fields and curvatures of these open NURBS curves were expressed by their control points. The relationships between control points were expressed when given two open NURBS curves occurred as Bertrand curve pairs at the points t = t(d), t = t(m-d), and arbitrary point in domain of these curves.