Optimum sample point selection algorithm for interpolating reflector surfaces with a prescribed accuracy

When using numerically defined reflector surfaces, it is often necessary to select a suitable number, and corresponding location, for the sample points used to interpolate the surface. Preferably this selection should be made in such a way that a minimum number of sample points is used, while maintaining a prescribed interpolation accuracy. This article describes a scheme for performing this task in a numerically efficient manner. The method starts from a large set of sample points /spl Xi//sub T/ which accurately describe the surface, a very small subset /spl Xi//sub 1/ of these points, a desired interpolation accuracy, /spl delta/, and a suitable interpolation scheme. Sample points are then judiciously transferred from /spl Xi//sub T/ to /spl Xi//sub 1/ while the changes in the interpolation accuracy are monitored. This procedure ultimately yields an optimum set /spl Xi//sub 0/ capable of interpolating the surface with the desired accuracy, /spl delta/. A key factor here is that, as the iterations progress, sample points are added only in regions where the error is large, thereby optimizing the sample point locations. To allow for this, the method requires an interpolator capable of handling non-uniform point locations. The interpolation scheme based on plate deformation theory has been selected for this task. This interpolator is very attractive since, in addition to handling non-uniform sample points, it yields excellent interpolation accuracy and provides continuous derivatives of all orders.