Specified–Precision Computation of Curve/Curve Bisectors

Abstract

The bisector of two plane curve segments (other than lines and circles) has, in general, no simple — i.e., rational — parameterization, and must therefore be approximated by the interpolation of discrete data. A procedure for computing ordered sequences of point/tangent/curvature data along the bisectors of polynomial or rational plane curves is described, with special emphasis on (i) the identification of singularities (tangent–discontinuities) of the bisector; (ii) capturing the exact rational form of those portions of the bisector with a terminal footpoint on one curve; and (iii) geometrical criteria the characterize extrema of the distance error for interpolants to the discretely–sample data. G 1 piecewise– parabolic and G 2 piecewise–cubic approximations (with O(h 4 ) and O(h 6 ) convergence) are described which, used in adaptive schemes governed by the exact error measure, can be made to satisfy any prescribed geometrical tolerance.