Illuminance critical points on generic smooth surfaces

Abstract

We consider the qualitative structure of the illuminance distribution that is due to the illumination of a generic smooth surface under the following simplifying conditions: The surface bidirectional reflection distribution function is a constant (that is, it is Lambertian and with constant albedo), the surface is homogeneously illuminated, and vignetting and interreflection effects are absent. In that case the critical points of the illuminance distribution are determined by the local third-order differential structure of the surface, that is, by the gradients of its principal curvatures. We find the classes of local surface structure that may yield various types of critical points under variation of the direction of incidence. The parabolic points of the surface play a distinguished role. Two distinct generic types of parabolic points exist that behave qualitatively differently with respect to illumination. Finally, we consider various relaxations of the rather restrictive initial constraints.