I have read that a Dalitz plot is nothing but the plot of $m_{12}^2$ vs $m_{23}^2$, and that the dots correspond to "events". However, this doesn't really tell me anything. My question can boil down to exactly what phase space variable the dots represent. I am currently trying to make a Dalitz plot for the three body decay $K_{long}\to 3\pi^0$. Now, I am able to get the boundary for the Dalitz plot by plotting the constraint
$$\cos(\theta_{12})=\frac{\frac{\left({m_\ell}^2+{m_\pi}^2-{m_{23}^2}\right) \left(-2 {m_\pi}^2+{m_{12}^2}+{m_{23}^2}\right)}{2 {m_\ell}^2}+2 {m_\pi}^2-{m_{12}^2}}{\sqrt{\frac{\left({m_\ell}^2+{m_\pi}^2-{m_{23}^2}\right)^2}{4 {m_\ell}^2}-{m_\pi}^2} \sqrt{\frac{\left(-2 {m_\pi}^2+{m_{12}^2}+{m_{23}^2}\right)^2}{{m_\ell}^2}-4 {m_\pi}^2}} $$
in the $m_{12}^2$ $m_{23}^2$ plane. However, I'm in the dark as to how to populate the inner region. My goal is to be able to answer the following question. Given the constraint above, how do I, for a given ($m_{12}^2$ ,$m_{23}^2$) know whether or not that corresponds to placing a dot there or not.
