  The spaces of point configurations on the projective line up to the action of
$\mathrm{SL}(2,\mathbb K)$ and its maximal torus are canonically compactified
by the Grothdieck-Knudsen and Losev-Manin moduli spaces $\overline M_{0,n}$ and
$\overline L_n$ respectively. We examine the configuration space up to the
action of the maximal unipotent group $\mathbb G_a\subseteq
\mathrm{SL}(2,\mathbb K)$ and define an analogous compactification. For this we
first assign a canonical quotient to the action of a unipotent group on a
projective variety. Moreover, we show that similar to $\overline M_{0,n}$ and
$\overline L_n$ this quotient arises in a sequence of blow-ups from a product
of projective spaces.
