To every $l\in{\cal L}_{in}$ we assign a pair $\left(f_{l}(0),sgn\left(\alpha_{l}(0)\right)\right)\in B^{sing}\times\{\pm 1\}$, where $sgn(\alpha_{l}(0))$ is a choice of sign in $\pm f_{l}^{\ast}(dy)$ (see data d) in the previous subsection). In this way we obtain a map $r:{\cal L}_{in}\to B^{sing}\times\{\pm 1\}$.

Axiom 1.

Map $r$ is one-to-one.

Let $U\subset Y$ be a simply-connected domain, and line $l$ intersects $U$. Let $I\subset{\bf R}_{+}$ be an interval such that $f_{l}(I)\subset U$. Then there exists a covariantly constant closed non-zero $1$-form $\beta_{U}$ in $U$ (with constant integer coefficients), such that $f_{l}^{\ast}(\beta_{U})=\alpha_{l}$, when both sides are restricted to $I$.

Axiom 2.

For any $t_{1},t_{2}\in I$ one has

$$\int_{f_{l}(t_{1})}^{f_{l}(t_{2})}\beta_{U}=t_{2}-t_{1}\,\,.$$

Let $l_{1},l_{2}\in{\cal L}$, $t_{1},t_{2}>0$ satisfy the condition $f_{l_{1}}(t_{1})=f_{l_{2}}(t_{2})=x\in Y$. In this case we say that lines $l_{1}$ and $l_{2}$ have a collision at $x$ at the times $t_{1}$ and $t_{2}$ respectively.

Axiom 3.

Under the above assumptions there are only two possibilities:

3a) either $l_{1}=l_{2}$ and $t_{1}=t_{2}$, or

3b) covector $\alpha_{l_{1}}(t_{1})$ is not proportional to $\alpha_{l_{2}}(t_{2})$. Then we may assume that $\alpha_{l_{1}}(t_{1})\wedge\alpha_{l_{2}}(t_{2})>0$. Under these conditions we require that for any coprime positive integers $n_{1},n_{2}$ there exists a unique line $l\in{\cal L}$ such that $l_{1}=p_{left(l)},l_{2}=p_{right}(l)$, $f_{l}(0)=x$ and $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$.

In other words, $l_{1}$ and $l_{2}$ are “parents of $l$”, and the direction covector of $l$ at the intersection point is a primitive integral linear combination of those for $l_{1}$ and $l_{2}$ (see Figure 4).

Axiom 4.

For every line $l\in{\cal L}_{com}$ there exist $l_{1}$ and $l_{2}$ such that they satisfy the condition 3b).

Axiom 5.

For any $x\in Y$ there are no more than two pairs $(l,t)\in{\cal L}\times(0,+\infty)$ such that $x=f_{l}(t)$. In other words, there are no more than two lines intersecting at a point in $Y$.

Let $l_{1},l_{2},t_{1},t_{2},x$ mean the same as in the Axiom 3, and assume that $\alpha_{l_{1}}(t_{1})\wedge\alpha_{l_{2}}(t_{2})>0$. Let us consider the set ${\cal L}_{(x)}$ of germs of all $l\in{\cal L}_{com}$ starting at $x$ (i.e. such that $f_{l}(0)=x$).

Axiom 6.

For any finite subset ${\cal L}^{\prime}\subset{\cal L}_{(x)}$ there is an orientation preserving homeomorphism of a neighborhood of $x$ onto a neighborhood of $(0,0)\in{\bf R}^{2}$ such that:

6a) Germs of oriented curves which are images of $l_{1}$ and $l_{2}$ get transformed into the germs at $(0,0)$ of coordinate axes $(x,0)$ and $(0,y)$ respectively.

6b) Germ of the image of $l\in{\cal L}^{\prime}$ gets transformed into the germ of the ray $\{(n_{1}t,n_{2}t)\,|\,t>0\}$ where $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$.

Figure 5 illustrates this axiom.

Axiom 7.

Let $p_{i}$ denotes either $p_{left}$ or $p_{right}$. Then for any $l\in{\cal L}$ there exists $N\geq 1$ such that if the line $p_{1}(p_{2}(\dots p_{N}(l)\dots)$ is well-defined then it belongs to ${\cal L}_{in}$.

This axiom says that any composed line $l\in{\cal L}_{com}$ appears as a result of finitely many collisions. The tree of ancestors of a given line form a tree embedded in $B$, see Figure 6.