Conic Approximation of Conic Offsets

AbstractA scheme for error-boundedG1conic approximation of offsets to conic Bézier segments is presented. The initial step involves segmentation of the conic at points inducing cusps on the offset, ensuring that each subsegment has a smooth offset amenable to conic approximation. The approximation scheme is based on interpolating the end points and tangents, and parametric midpoint, of the exact offset. Conditions for the existence of real interpolants to such data are derived, and a sufficient condition for unambiguous definition of a distance function between the conic and its approximate offset is identified in terms of the control polygon. It is shown that extrema of this function occur along common normal lines to the conic and approximate offset, a condition satisfied at the real roots on (0,1) of a degree 10 polynomial in the conic parameter. These roots may be isolated and approximated using the Bernstein form, yielding a sharp geometric bound on the approximation error. Under subdivision the scheme produces rapidly converging approximations, satisfying any desired accuracy, for precision engineering applications