Planar curve offset based on circle approximation

Abstract

An algorithm is presented to approximate planar offset curves within an arbitrary tolerance ϵ > 0. Given a planar parametric curve C( t) and an offset radius r, the circle of radius r is first approximated by piecewise quadratic Bézier curve segments within the tolerance ϵ. The exact offset curve C r ( t) is then approximated by the convolution of C( t) with the quadratic Bézier curve segments. For a polynomial curve C( t) of degree d, the offset curve C r ( t) is approximated by planar rational curves, C r a ( t)s, of degree 3 d −2. For a rational curve C( t) of degree d, the offset curve is approximated by rational curves of degree 5 d −4. When they have no self-intersections, the approximated offset curves, C r a ( t)s, are guaranteed to be within ϵ-distance from the exact offset curve C r ( t). The effectiveness of this approximation technique is demonstrated in the offset computation of planar curved objects bounded by polynomial/ rational parametric curves.