Four-bar linkage (the problem)
This activity belongs to the GeoGebra book Linkages.
Let us now analyze the particular case (and habitual from now on) of considering all the bars of equal length.
In the following construction we can observe the previous case of 4 points and 4 bars (a). Now the figure that is formed can be either the rhombus UOEF (or UOEF', depending on the position of E) or the set of double bars UF'=UO, EF'=EO (or UF=UO, EF=EO depending on the position of E). In any case, when fixing E, let us observe that although the construction is locally rigid (as we have already seen in the general case), there is still no global rigidity, since the rhombus is not congruent with the set of two double bars.
Furthermore, in [1] we can read:
"On the other hand, modeling linkages with dynamic geometry poses other kind of challenges. For instance, it is difficult to model a four-bar planar linkage where all vertices behave similarly, that is, showing in a similar manner the degrees of freedom of the flexible parallelogram when one drags any one of the vertices. Let’s fix two contiguous vertices, say, O and U and consider only the internal degrees of freedom.
Then the two remaining vertices, F, E, should have each one degree of freedom, but not simultaneously. Dragging F, point E should move, and vice versa. But a dynamic geometry construction tends to assign the shared degree of freedom to just one of them, depending on the construction sequence, and not to the other. Typically, if F is constructed first, when we can drag it, E will move; but we can not drag E."
Although from a static point of view, this geometric model clearly looks like a four-bar rhombus, it does not have a visual dynamic behavior as one would expect. We are not referring to the physical object, whose characteristics and laws (kinematic behavior, inertia, tension, etc.) have not been mathematically modeled, but to the purely geometric characteristics of the physical model, which there is no problem modeling algebraically (with GeoGebra) in form of coordinates and equations.
Therefore, this geometric model results, in its dynamic behavior, poorer than its algebraic counterpart.
Author of the construction of GeoGebra: Rafael Losada