Combinatorial decompositions and homogeneous geometrical processes

This paper considers line processes and random mosaics. The processes are assumed invariant with respect to the group of translations ofR ². An expression for the probabilities ,k=0, 1, 2,... to havek hits on an interval of lengtht taken on a typical line of direction (the hits are produced by other lines of the process) is obtained. Also, the distribution of a length of a typical edge having direction in terms of the process {P _i , _i } is found, hereP _i is the point process of intersections of edges of the mosaic with a fixed line of direction and the mark _i is the intersection angle atP _i . The method is based on the results of combinatorial integral geometry.