New Approximation Methods for Planar Offset and Convolution Curves

Abstract

Abstract. We present new methods to approximate the offset and convolution of planar curves. These methods can be used as fundamental tools in various geometric applications such as NC machining and collision detection of planar curved objects. Using quadratic curve approximation and tangent field matching, the offset and convolution curves can be approximated by polynomial or rational curves within the tolerance of approximation error > 0. We suggest three methods of offset approximation, all of which allow simple error analysis and at the same time provide high-precision approximation. Two methods of convolution approximation are also suggested that approximate convolution curves with polynomial or rational curves.

Polynomial/Rational Approximation of Minkowski Sum Boundary Curves

“…(5) may not have a unique rational solution of s(t) when the curve C 2 (s) has degree higher than two. A simple way of approximating the convolution curve (C 1 * C 2 )(t) with a rational curve is to approximate either (i) the curve C 2 (s) by a quadratic polynomial curve [30,31], or (ii) the reparametrization s(t) by a rational function [31]. Lee et al [30] applied the first approach to the planar curve offset problem in which the curve C 2 (s) is an exact circle.…”

“…Therefore, Lee et al [30] approximated the offset circle by a sequence of quadratic polynomial Bézier curves and approximated the offset curve by computing the convolution curves of the given base curve and the Bézier curves approximating the offset circle. Lee et al [31] suggested convolution curve approximation methods based on the two approaches (i) and (ii) above and compared their experimental results.…”

“…In Section 4, the methods of Lee et al [31] are classified according to approaches based on (i) quadratic curve approximation, (ii) reparametrization, and (iii) tangent field matching. Section 4 also presents other methods corresponding to approaches based on (iv) control polygon and (v) interpolation.…”

Offset approximation based on reparameterizing the path of a moving point along the base circle

“…Taking a different approach, Lee, et al [14][15][16] proposed to treat the offset as a convolution problem. They regarded the offset curve as a convolution of a sweeping circle moving along the base curve whose radius was equal to the offset radius.…”

“…Method CAO [14] approximates the sweeping circle with piecewise quadratic polynomial Bézier curves, and takes the convolution of the circle and the base curve as the offset approximation. Methods LRC, TMC and SRC [15][16] represent the sweeping circle with piecewise quadratic rational Bézier curves, and take the convolution of the reparametrized sweeping circle and the base curve as the approximation. Method CAMO [16][17] approximates the circle with polynomial curves, and then reparameterizes the convolution of the circle and the base curve.…”