Sharp, Quantitative Bounds on the Distance Between a Bezier Curve and its Control Polygon

The distance between a Be-....ier segment and its control polygon is bounded in terms of the second differences of the control point sequence and a constant that depends only on the degree of the polynomial. The constant derived here is the smallest possible and is sharp for the Hausdorff distance between control polygon and curve segment. The bound provides a straightforward proof of quadratic convergence of the sequence of control polygons to the Bczicr segment under subdivision or degree-fold degree-raising and establishes the explicit convergence constants. The bOl1.Ild also allows analyzing the optimal choice of subdivision parameter for adaptive refinement of quadratic and cubic segments and it may be useful to establish better bounding regions. 1 Curved geometry and control polygons A Widely used, efficient, intuitive way to specify, represent and reason about curved, nonlinear geometry for design and modeling is the control point or control polyline paradigm: for popular representations like the B-spline and the Bernstein·Bezier representation the curve-shape is outlined by the broken line connecting the control points. For many applications, e.g. rendering, intersection testing, design, this raises the question just lJOw well the control line approximates the exact curved geometry. This paper gives a simple, optimal quantitative answer to the question in terms of the second differences of the control point sequence of the Bezier representation and a constant that depends only on the degree of the polynomial. In these terms the bound is generically sharp, i.e. there exist commonly used curves such that the bound is taken OD and any reduction of the constant would not yield a bound. Remarkably, the bound remains sharp under degree-raising and subdivision, i.e. refinement of the piecewise linear control structure to better approximate the curved geometry. This yields for example a sharp a priori • User Serviccs, University of Delaware tSupported by NSF NYI grant 9157B06·CCR ~Computer Scienccs, Purdue University