Some integral equations in geometrical probability

Abstract

SUMMARY The 'point quadrat' is a sharp needle inserted into vegetation, contacts between needle and foliage surfaces being counted. Refinement of established botanical techniques, based on the absence of preferred compass orientation, leads to three integral equations. Equation (A) relates the variation of contact frequency with quadrat angle, f(fl), to the distribution of foliage density with foliage angle, g(ca). Equation (B), applicable to stem-like organs, relates f(,8) to the distribution of foliage (stem surface) density with axial angle, h(y); and (C) connects g(a) and h(y). The kernel of (A) is known, and those of (B) and (C) are found from geometrical considerations. The solution of (C) is found and combined with that of (A) to yield that of (B). The kernel of (B) involves all three complete elliptic integrals. Three identities involving elliptic integrals follow. A trio of equations analogous to (A), (B), (C) holds for any class of axisymmetrical organs of which the members are geometrically similar. The utility of these equations in practice depends on the differential order of their solutions: the higher the order, the greater the amplification of errors. The order is 21 for (A); and 3 and 1 for (B) and (C) in the 'stem' case. Similar problems can be formulated in other spaces. An example is the problem of estimating from quadrat contact observations the density of curves drawn on a plane and its distribution with respect to curve direction. This leads to an equation analogous to (A). The solution is of differential order 2.