a)

A compact oriented surface $B$, a finite subset $B^{sing}\subset B$.

b)

A ${\bf Z}$-affine structure on $Y=B^{sm}=B\setminus B^{sing}$ with the standard singularities near each $b\in B^{sing}$.

c)

A set ${\cal L}$ of lines. With each line $l\in{\cal L}$ there is an associated continuous map $f_{l}:(0,+\infty)\to Y$. We assume that ${\cal L}$ is decomposed into a disjoint union of two subsets ${\cal L}={\cal L}_{in}\sqcup{\cal L}_{com}$. Lines belonging to ${\cal L}_{in}$ are called initial, while those in ${\cal L}_{com}$ are called composite. We assume that for any $l\in{\cal L}$ there exists a continuous extension $f_{l}:[0,+\infty)\to B$ such that $f_{l}(0)\in B^{sing}$ if $l\in{\cal L}_{in}$ and $f_{l}(0)\in Y=B^{sm}$ if $l\in{\cal L}_{com}$.

d)

A collection of covariantly constant nowhere vanishing integer-valued $1$-forms $\alpha_{l}\in\Gamma((0,+\infty),f_{l}^{\ast}((T^{\ast})^{\bf Z}),l\in{\cal L}$. We assume that for $l\in{\cal L}_{in}$ in the standard coordinates $(x,y)$ near singular point $f_{l}(0)$ we have: $f_{l}(t)=(0,t)$ or $f_{l}(t)=(0,-t)$ for all sufficiently small $t>0$, and $\alpha_{l}(t)=\pm f_{l}^{\ast}(dy)$.

e)

A map ${\cal L}\to{\cal L}\times{\cal L},\,\,\,l\mapsto(p_{left}(l),p_{right}(l))$ (the letter $p$ stands for “parent”: one can think about these lines as “generating $l$ in a collision”).

Notice that since the form $dy$ is invariant with respect to the monodromy, the condition in d) is coordinate-independent. The covector $\alpha_{l}(t)$ will be called a direction covector of $l$ at time $t$. It gives rise to a half-plane

$$P_{l,t}^{(0)}=\{v\in T_{f_{l}(t)}Y|\langle\alpha_{l}(t),v\rangle>0\}\,\,.$$