Real--time CNC interpolators for Bezier conics

Arbitrary conic segments can be specified in the rational Bézier form, r(ξ ) for ξ ∈ [0, 1], by control points p0,p1,p2 and a scalar weight w1. An expression for the cumulative arc length function s(ξ), amenable to accurate and efficient evaluation, is required in formulating real-time CNC interpolators capable of achieving a desired (constant or varying) feedrate V = ds/dt along such curves. For w1 = 1 (a parabola), s(ξ) admits a closed-form expression that entails a single square root and natural logarithm in its evaluation. However, for w1 < 1 (an ellipse) or w1 > 1 (a hyperbola), complete and incomplete elliptic integrals of the first and second kind arise in s(ξ). A recursive algorithm, based on the arithmetic-geometric mean, provides a rapidly-convergent scheme to compute such integrals to machine precision in real-time applications. These methods endow CNC machines with the ability to realize time-dependent feedrates precisely along “simple” analytic curves (conics), furnishing a natural complement to the currently-available exact real-time interpolators for free-form Pythagorean-hodograph (PH) curves.