Loedel diagrams are special cases of the Minkowski spacetime diagram and are very easy to use when only two inertial frames are considered. It is an excellent tool for teaching Special Relativity, as first discussed in 1957 by Enrique Loedel Palumbo (1901-1962), a Latin American physicist.
Basic Loedel diagrams picture two inertial frames in relative motion in a symmetrical fashion around an imaginary central Minkowski reference frame. The two inertial frames are shown in blue and red in Figure 1. Note that the blue and red coordinate systems both have skewed axes, but that the blue time axis is normal to the red space axis and vice-versa.
Also note that if v approaches c, the angle phi will approach 90 degrees, meaning that the axes of the blue frame will coincide with the lightcone- line and the red frame's axes will coincide with the lightcone+ line. When v = 0, the two coordinate systems will obviously coincide and we have the Minkowski coordinate system.
One of the benefits of the diagram is that unlike in Minkowski diagrams, the scales of both axes of both frames are identical. In effect, the worldlines of both observers progress along their respective time axes at the same rate. Hence, the Loedel diagram does not give apparent preference to one of the inertial frames (i.e., the one chosen as reference frame). In Fig. 1 the bullets indicate the identical scales of the two time axes.
The real purpose of these diagrams is to graphically show the Lorentz transformation between two inertial frames, as in Figure 2. The event E has different coordinates in the two inertial frames, but the two frames will agree on the spacetime interval S. We read off the coordinates of the event by constructing (dotted) lines parallel to the respective (skewed) axes-systems.
Proof that the diagram indeed preserves the spacetime interval (S) between two inertial frames in relative motion is easily seen from Figure 2. The line L is a common hypotenuse of two normal triangles (OPQ and EPQ) and hence the spacetime interval (S) is the same as measured in either frame.
As a special case of the Minkowski diagram, the preservation of the scales of the two axis-systems is the major significant benefit. It is possible to add a third inertial frame, but in such a case the Minkowski diagram is easier to work with.