Arclength Approximation for Reference Point Generation

In applications such as CNC machining highway and railway design manufacturing industry and animation there is an ongoing need to systematically generate sets of reference points with prescribed arclengths along parametric curves with considerable accuracy and realtime performance. In the process mechanisms to result in a parameter set that yield the coordinates of the reference points along the curve Q(t) = {x(t), y(t)} are sought. The presence of arclength parameterizable expressions usually yields the parameter set that is necessary to generate the reference points however for typical design curves such an expression is often not available in closed form. It is desirable to find efficient ways to compensate for lack of arclength parameterization. In this book, several analytical and numerical methods for approximating arclength parameterization are presented. These methods are examined for both accuracy and realtime processing requirements. The application of generating uniformly spaced reference points along the paths of several curves is chosen for the purposes of illustration and bench marking among the presented methods.