Consider a three-point scattering process of the form
$$ \Phi_{1}(p_{1}) + \Phi_{2}(p_{2}) \longrightarrow \Phi_{3}(p_{3}). $$
You have a conservation constraint,
$$ p_{1} + p_{2} = p_{3}; $$
and three massive on-shell constraints:
$$ m_{i}^{2} = -\left\Vert p_{i} \right\Vert^{2}. $$
From the conservation constraint, you find three relations
$$ p_{1} \cdot p_{2} - m_{1}^{2} = p_{3} \cdot p_{1}, $$
$$ p_{1} \cdot p_{2} - m_{2}^{2} = p_{2} \cdot p_{3}, $$
$$ p_{3} \cdot p_{1} + p_{2} \cdot p_{3} = -m_{3}^{2}. $$
Solving these yields
$$ p_{1} \cdot p_{2} = \frac{m_{1}^{2} + m_{2}^{2} - m_{3}^{2}}{2}, $$
$$ p_{2} \cdot p_{3} = \frac{m_{1}^{2} - m_{2}^{2} - m_{3}^{2}}{2}, $$
$$ p_{3} \cdot p_{1} = \frac{m_{2}^{2} - m_{3}^{2} - m_{1}^{2}}{2}. $$
Next you introduce (null) Sudakov vectors:
$$ p_{1} = n_{1} + \frac{m_{1}^{2}}{S_{12}} n_{2}, $$
$$ p_{2} = n_{2} - \frac{m_{2}^{2}}{T_{23}} n_{3}, $$
$$ p_{3} = n_{3} - \frac{m_{3}^{2}}{T_{31}} n_{1}. $$
Here I have also introduced three 2-Sudakov invariants:
$$ S_{12} \equiv -\left\Vert n_{1} + n_{2} \right\Vert^{2} = - 2 \left( n_{1} \cdot n_{2} \right), $$
$$ T_{23} \equiv -\left\Vert n_{2} - n_{3} \right\Vert^{2} = 2 \left( n_{2} \cdot n_{3} \right), $$
$$ T_{31} \equiv -\left\Vert n_{3} - n_{1} \right\Vert^{2} = 2 \left( n_{3} \cdot n_{1} \right). $$
In terms of the 2-Sudakov invariants, you have
$$ m_{3}^{2} - m_{1}^{2} - m_{2}^{2} = S_{12} + m_{2}^{2} \left( \frac{T_{31}}{T_{23}} \right) + \frac{m_{1}^{2} m_{2}^{2}}{S_{12}}, $$
$$ m_{1}^{2} - m_{2}^{2} - m_{3}^{2} = T_{23} + m_{3}^{2} \left( \frac{S_{12}}{T_{31}} \right) + \frac{m_{3}^{2} m_{2}^{2}}{T_{23}}, $$
$$ m_{2}^{2} - m_{3}^{2} - m_{1}^{2} = T_{31} + m_{1}^{2} \left( \frac{T_{23}}{S_{12}} \right) + \frac{m_{3}^{2} m_{1}^{2}}{T_{31}}.$$
Now you introduce Regge-Mandelstam invariants,
$$ r_{12} \equiv \frac{m_{3}^{2} - m_{1}^{2} - m_{2}^{2}}{2 m_{1} m_{2}}, $$
$$ r_{23} \equiv \frac{m_{2}^{2} + m_{3}^{2} - m_{1}^{2}}{2 m_{2} m_{3}}, $$
$$ r_{31} \equiv \frac{m_{3}^{2} + m_{1}^{2} - m_{2}^{2}}{2 m_{1} m_{3}}; $$
and Regge-Sudakov invariants,
$$ R_{12} \equiv \frac{S_{12}}{m_{1} m_{2}}, $$
$$ R_{23} \equiv -\frac{T_{23}}{m_{2} m_{3}}, $$
$$ R_{31} \equiv -\frac{T_{31}}{m_{1} m_{3}}. $$
The relations above become
$$ 2 r_{12} = R_{12} + \frac{R_{31}}{R_{23}} + \frac{1}{R_{12}}, $$
$$ 2 r_{23} = R_{23} + \frac{R_{12}}{R_{31}} + \frac{1}{R_{23}}, $$
$$ 2 r_{31} = R_{31} + \frac{R_{23}}{R_{12}} + \frac{1}{R_{31}}. $$
The solution to these equations is very complicated. Furthermore, these equations appear in the null decomposition of three independent massive bodies. Using the simpler version above, in terms of the 2-Sudakov invariants, the solution is
$$ S_{12} = -m_{1}^{2}, $$
$$ T_{23} = -m_{2}^{2}, $$
$$ T_{31} = -m_{3}^{2}. $$
Thus,
$$ R_{12} = - \frac{m_{1}}{m_{2}}, $$
$$ R_{23} = \frac{m_{2}}{m_{3}}, $$
$$ R_{31} = \frac{m_{3}}{m_{1}} .$$
It follows that,
$$ p_{1} = n_{1} - n_{2}, \qquad p_{2} = n_{2} + n_{3}, \qquad p_{3} = n_{3} + n_{1}.$$
That is, the three massive momenta can be written as very simple combinations of Sudakov vectors.
