Figure 3
This app represents an interactive version of figure 3.
This figure is exactly the same as figure 2 for the bismar scale, but in this case we have no physical rod, which is substituted by a black segment on the (horizontal) velocity axis.
Move the point pf defining the position of the momentum of the outgoing particle (this drags the fulcrum F on the spaceteime scale) and see how the mass M changes.
The mass of the particle m (initially at rest) can be changed moving the green point.
You can also change the incoming velocity (blue point vi on the horizontal axis).
Figure 4
This app represents an interactive version of figure 4.
Comparing with the classic case (iigure 3) we see that now the horizontal rod has been substituted by an arc of a hyperbola. It is called the "mass shell" and defines the possible momentum/energy values for the test particle.
Move the point pf defining the position of the momentum of the outgoing particle (this drags the fulcrum F on the relativistic spaceteime scale) and see how the mass M changes.
Both the mass of the test particle m (green point) and its initial momentum (blue point pi) can be changed too. Observe that the point I lies always on the mass shell (black hyperbola).