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We define the length of the particle as the length of the longest géodésie arc within the particle (see fig. 3-3). L(X) = sup d (x,y) X x,y£x This définition may seem somewhat arbitrary to the reader. . However, it has three big advantages : i) this définition is général, as it is applicable to particles of any shape. ii) this définition is robust, in the sense that a slight déformation of a particle slightly modifies its length. iii) this définition is operational, insofar as a technology exists or can be devised to use it. Sometimes, particles intersect themselves. A discussion of this is presented in Appendix 1. 3 ~ GEODESIC DISTANCE FUNCTION AND PARTICLE RECONSTRUCTION Now, let us assume that X is not a single particle, but a population of particles. If the two points x and y belong to two distinct particles, there is no path in X linking x and y, and we can write dx(x,y) = + « . It can easily be shown that the function d satisfies ail the X properties of a distance function : i) dx(x,y) ^ 0 and d (x9y) =0 if and only if x = y ii) dx(x,y) = dx(y,x) iii) dx(x,z) < dx(x,y) + d (y,z) In the following, the function d is termed "géodésie distance X function". The reader can compare on fig. 4-1 and 4-2 the disks B (x,X) X and B(x,X) with center x and radius X, with the géodésie metric d and X