Fast and Accurate Parametric Curve Length Computation

This paper describes a simple technique for evaluating the length of a parametric curve. The technique approximates sections of the curve to the arc of a circle using only sample points on the curve: no computation of derivatives or other properties is required. This method is almost as quick to implement and compute as chord-length summation but is much more accurate : it also provides recursion criteria for adaptive sampling. At the end of paper, we discuss briefly the way the algorithm extends to estimate the area of a parametric surface. Introduction The simplest, quickest, and perhaps most widely used technique for determining the length of a parametric curve is chord-length summation (1). Other standard approaches include recursive summation such as the DeCastlejeu algorithm for Bézier curves (2), or integrating the arc length function either by some quadrature technique or by recasting the problem in the form of integrating a differential equation using a technique such as 4th-order Runge-Kutta. This paper presents a new technique that has the following properties: easy to implement. cheap to compute. good worst-case accuracy characteristics. applies to any parametric curve. requires only evaluation of points on the curve. The technique can be summarized as follows: Given points precomputed along the curve Approximate successive triples of points by circular arcs, the lengths of which can be computed cheaply For the basic algorithm, sum the length of non-overlapping arcs; alternatively, sum overlapping arcs to get a second length estimate Regions of high curvature can readily be detected and the algorithm made recursive as necessary. Test results show that the accuracy of the method increases as the 4th power of the number of points evaluated: so doubling the work will result in a 16-fold reduction in the error. Typically this means that evaluating 50 points along the curve will result in an error of a few parts in 108, representing an improvement of 3 orders of magnitude over that obtained by chord-length summation of distances between the same sample points along the curve. Estimation of arc length Consider three points along a curve : P0, P1, and P2. A circle with radius r and center C can be fitted to these points. Let D1 be the distance from P0 to P2, and D2 be the sum of the distances from P0 to P1 and from P1 to P2. P0