The aim of this paper is to develop a straightforward rigorous and flexible computational method to determine the coordinate points on an aspheric surface. The computational method chosen is based on the basic slope-point form of a straight-line equation [slope-point method (SPM)]. The practical instrumental example chosen to illustrate this method is a rotationally symmetric catadioptric collimator for a light-emitting diode (LED) source. This optical system has both a refractive and a totally internally reflective aspheric surface. It is a particularly illuminating example because it requires careful computational attention to the smooth transition between the refracting inner zones and the reflective outer zones of the aperture. The chosen SPM computational method deals satisfactorily with the transition points at the junction between the refractive and total internal reflecting (TIR) zones of the collimator. As part of this study, the effect of the position of the start point of the SPM surface evolution for the TIR zones of the collimator emerges as being particularly important, and the details of this are discussed. Finally, an extension of the basic SPM-based method is used to generalize the development of the catadioptric collimator surfaces to illustrate this general algorithm for aspheric surface design for an extended LED light source.
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