Hausdorff Distance between the Offset Curve of Quadratic Bezier Curve and Its Quadratic Approximation

In this paper, we present the exact Hausdorff distance between the offset curve of quadratic Bézier curve and its quadratic GC1 approximation. To illustrate the formula for the Hausdorff distance, we give an example of the quadratic GC1 approximation of the offset curve of a quadratic Bézier curve. 1. Preliminaries Quadratic Bézier curves and their offset curves are widely used in CAD/CAM or Computer Graphics. But offset curve of quadratic Bézier curve cannot be expressed in Bézier form. In the recent twenty years, many papers about the quadratic Bézier approximation [2, 7, 13] or the offset curve [6, 8, 10, 11, 15] have been published. Especially as the error measurement between the target curve and approximation curve, the Hausdorff distance is used in CAD/CAM or Approximation Theory. The Hausdorff distance dH(p,q) between two curves p(s), s ∈ [a, b] and q(t), t ∈ [c, d], is given by dH(p,q) = max{ max s∈[a,b] min t∈[c,d] |p(s)− q(t)|, max t∈[c,d] min s∈[a,b] |p(s)− q(t)|}. (For more knowledge about the Hausdorff distance, refer to [1, 14, 12]) By the way, it is not easy to find the Hausdorff distance between planar curve and quadratic Bézier curve. The Hausdorff distance is obtained from the maximum distance between p(s0) and q(t0) satisfying p′(s0) ‖ q′(t0) and p′(s0)⊥ −−−−−−−→ p(s0)q(t0) (1.1) when they have the same end points, as shown in Figure 1. Thus to get the Hausdorff distance it requires to solve nonlinear system of two variables such as in Equation (1.1). Received July 11, 2007. 2000 Mathematics Subject Classification. 41A05, 41A15, 65D05, 65D07, 65D17.