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FLATNESS AND R0UGHNE3S OF NON-PLAUAR SURFACES J. SERRA June 19|S2 1 - INTRODUCTION The purpose of this note is to provide a description of the irregularity of non planar surfaces. A few requirements will be taken into account : i) In practical metallography, non planar surfaces such as fractures, are accessed by vertical profiles. The approach is non destructive, but emphasizes the vertical sections (in différent directions). ii) We want the descriptors to be very easily computed. There- fore, we will restrict ourselves to the cross sections by random straight lines of the material under study. iii) We wish the descriptors to possess a stereological rnean- ing, i.e. to be interpreted in terms of characteristics of the surface in the 3-D space. iv) in addition, we would like to extend the approach to grey tones functions in two and three dimensions, where the gra- nularity of the textures is a notion similar to the roughness of a surface. Some interesting studies have been made on the subject and presented in particular at the 3rd European Congress for Stereology (Réf. 1). K. Wright and B. Karlsson (réf. 1, pp. 247-253) have adop- ted the same prerequisites as we did, but curiously, they avoiaed Crofton's formula to close the gap between intercepts and surface area. They preferred to resort to a polygonal model of the surface, the necessity of which is problematic. I). and P. Jeulin have pro- posed clever stochastic models which seem more appropriate than the classical Gaussian surfaces generally used for such struc¬ tures. Finally, M. Coster and A. Deschanvres (réf. 2, pp. 61-73) consider the fracture as a fractal surface. These last two approa- ches seem promising, but are both far beyond the limited goal we