The structure and explicit determination of convex-polygonally generated shape-densities

This paper is concerned with the shape-density for a random triangle whose vertices are randomly labelled and i.i.d.-uniform in a compact convex polygon K. In earlier work we have already shown that there is a network of curves (the singular tessellation T(K)) across which suffers discontinuities of form. In two papers which will appear in parallel with this, Hui-lin Le finds explicit formulae for (i) the form of within the basic tile T0 of T(K), and (ii) the jump-functions which link the local forms of on either side of any curve separating two tiles. Here we exploit these calculations to find in the most general case. We describe the geometry of T(K), we examine the real-analytic structure of within a tile, and we establish by analytic continuation an explicit formula giving in an arbitrary tile T as the sum of the basic-tile function and the members of a finite sequence of jump-functions along a .stepping-stone' tile-to-tile route from T0 to T. Finally we comment on some of the problems that arise in the use of this formula in studies relating to the applications in archaeology and astronomy.