  We consider scattering amplitudes in planar N = 4 supersymmetric Yang-Mills
theory in special kinematics where all external four-dimensional momenta are
restricted to a (1+1)-dimensional subspace. The amplitudes are known to satisfy
non-trivial factorisation properties arising from multi-collinear limits, which
we further study here. We are able to find a general solution to these
multi-collinear limits. This results in a simple formula which represents an
n-point superamplitude in terms of a linear combination of functions S_m which
are constrained to vanish in all appropriate multi-collinear limits. These
collinear-vanishing building blocks, S_m, are dual-conformally-invariant
functions which depend on the reduced m-point kinematics with 8 \leq m \leq 4l.
For MHV amplitudes they can be constructed directly using, for example, the
approach in Ref. [1]. This procedure provides a universal uplift of lower-point
collinearly vanishing building blocks S_m to all higher-point amplitudes. It
works at any loop-level l \geq 1 and for any MHV or N^kMHV amplitude. We
compare this with explicit examples involving n-point MHV amplitudes at 2-loops
and 10-point MHV amplitudes at 3-loops. Tree-level superamplitudes have
different properties and are treated separately from loop-level amplitudes in
our approach. To illustrate this we derive an expression for n-point tree-level
NMHV amplitudes in special kinematics.
