Shape-preserving Interpolation with Biarcs and NURBS

Abstract

Non-Uniform Rational B-Splines (NURBS) curve has acquired great significance in the field of Computer Aided Design and Machining due to their ability to draw a large variety of shapes in an interactive computer graphics environment. A biarc curve is a composition of two circular arcs such that they are tangent continuous at the point of join. Biarcs have replaced traditionally used line segments in approximating curves and surfaces for generating tool paths of Computerised cutting machines called CNC (Computerised Numerical Controlled) machines. This is due to their ability to be at a greater proximity to the original curve with fewer number of segments. Since most of the machining tools can move only in straight lines and circular arcs, it is desirable that the tool paths be composed of biarcs and/or straight line segments. Shape preserving interpolation is a technique of drawing a curve through a set of points such that the shape represented by the data points are preserved. Both NURBS and biarc curves are not essentially shape preserving curves; however, if certain constraints are imposed on them, they are able to preserve the shape represented by the data points. This work proposes a technique that incorporates both NURBS and biarcs to perform the interpolation. The advantages are twofold; it acts as a common platform for the two techniques to operate together, which is novel, and the fitted NURBS curve can be approximated by biarcs, which has applications in the machining industry.