  We give an explicit geometric way to build mixed-integer programming (MIP)
formulations for unions of polyhedra. The construction is simply described in
terms of spanning hyperplanes in an r-dimensional linear space. The resulting
MIP formulation is ideal, and uses exactly r integer variables and 2 x (# of
spanning hyperplanes) general inequality constraints. We use this result to
derive novel logarithmic-sized ideal MIP formulations for discontinuous
piecewise linear functions and structures appearing in robotics and power
systems problems.
