Constructing acceleration continuous tool paths using Pythagorean Hodograph curves

Any set of C2 planar boundary data (two points with associated velocities and acceleration vectors) can be interpolated by a PH curve of degree 9. In the generic case there are four such curves [R.T. Farouki, J. Manjunathaiah, S. Jee, Design of rational cam profiles with Pythagorean Hodograph curves, Mech. Mach. Theory 33 (1998) 669–682]. In this paper we give a detailed description and a qualitative analysis of these solutions. In particular we label the four solutions and compare their fairness from the point of view of possible application. For this purpose we consider the Hermite data taken from an analytical curve and study the behavior of the solutions for decreasing step-size h. This allows us to identify explicitly the solution which—for sufficiently small step-size—matches the shape of the curve with a high precision (the approximation order is 6). Consequently we are able to develop a highly robust, fast and precise algorithm converting analytical curves or curves described in G-code into a C2 continuous PH spline curve of degree 9.