Curve interpolation based on the canonical arc length parametrization

We use the canonical equations (CE) of differential geometry, a local Taylor series representation of any smooth curve with parameter the arc length, as a unifying framework for the development of new CNC algorithms, capable of interpolating 2D and 3D curves, represented parametrically, implicitly or as surface intersections, with accurate feedrate control. We use a truncated form of the CE to compute a preliminary point, at an arc distance from the last interpolation point selected to achieve a desired feedrate profile. The next interpolation point is derived by projecting the preliminary point on the curve. The coefficients in the CE involve the curve’s curvature, torsion and their arc length derivatives. We provide computing procedures for them for common Cartesian representations, demonstrating the generality of the proposed method. In addition, our algorithms admit corrections, which render them more accurate in terms of the programmed feedrate, compared to existing parametric algorithms of the same order.