Clarification of the definition of coordinates used to simplify the n-particle Lorentz invariant phase space

Question

To simplify the Lorentz invariant phase space element

$$dPS = \frac{1}{(2\pi)^{3n}}\prod\limits_1^2\delta(p_{\mu}p^{\mu}-m^2)\Theta(p^0)d^4p$$

the relation

$$\delta(p_{\mu}p^{\mu}-m^2)\Theta(p^0)d^4p = \frac{p}{2}dE d\Omega = \frac{p_{\bot}}{2E}dp_{\parallel}dp_{\bot}d\phi$$

can be be applied.

While I see how the first equality is vaild, I fail having any intuition for visualizing the new coordinate system introduced in the second step.

Can anybody give some (visual) enlightment about this coordinate system which usese some kind of orthogonal and parallel momentum coordinates?  A corresponding figure would probably be helpful ...