  We perform the twistor (half-Fourier) transform of all tree n-particle
superamplitudes in N=4 SYM and show that it has a transparent geometric
interpretation. We find that the N^kMHV amplitude is supported on a set of
(2k+1) intersecting lines in twistor space and demonstrate that the
corresponding line moduli form a lightlike (2k+1)-gon in moduli space. This
polygon is triangulated into two kinds of lightlike triangles lying in
different planes. We formulate simple graphical rules for constructing the
triangulated polygons, from which the analytic expressions of the N^kMHV
amplitudes follow directly, both in twistor and in momentum space. We also
discuss the ordinary and dual conformal properties and the cancellation of
spurious singularities in twistor space.
