  We undertake a combinatorial study of the piecewise linear map g : R^{2m+2n}
--> R^{mn} which assigns to the four vectors a, A in R^m and b, B in R^n the m
by n matrix given by g_{ij} = min (a_i + b_j, A_i+B_j). This map arises
naturally in Pachter and Sturmfels's work on the tropical geometry of
statistical models. The image of g has been a subject of recent interest; it is
the positive part of the tropical algebraic variety which parameterizes
n-tuples of points on a tropical line in m-space.
  The domains of linearity of g are the regions of the real hyperplane
arrangement A_{m,n}, corresponding to the complete bipartite graph K_{m,n}. We
explain how the images of (some of) the regions provide two polyhedral
subdivisions of the image of g, one of which is a refinement of the other. The
finer subdivision is particularly nice enumeratively: it has 2 {m \choose 2} {n
\choose 2} r_{m-2,n-2} maximum-dimensional cells, where r_{m-2,n-2} is the
number of regions of the arrangement A_{m-2,n-2}.
