Abstract
Arbitrary conic segments can be specified in the rational Bézier form, r(ξ) for ξ∈[0,1], by control points p 0, p 1, p 2 and a scalar weight w 1. 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=d s/d t along such curves. For w 1=1 (a parabola), s( ξ) admits a closed-form expression that entails a single square root and natural logarithm in its evaluation. However, for w 1<1 (an ellipse) or w 1>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.
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