Real-time CNC interpolators for Bezier conics
Arbitrary conic segments can be specified in the rational Bezier form, r(xi) for xi is an element of [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(xi), 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 w(1) = 1 (a parabola), s(xi) admits a closed-form expression that entails a single square root and natural logarithm in its evaluation. However, for w(1)   1 (a hyperbola), complete and incomplete elliptic integrals of the first and second kind arise in s(). 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. (C) 2001 Elsevier Science B.V. All rights reserved.
elliptic integrals arithmetic-geometric mean cnc machines time-dependent feedrates real-time interpolators conic bezier curves