As we have seen, simple hinged rod mechanisms can solve both theoretical and practical problems. The simplest of these mechanisms is the articulated quadrilateral. If you consider the ABCD quadrilateral with a fixed side AD, it is clear that the position of one of the rods determines that of the remaining ones. This mechanism has only one degree of freedom.
If, in the version shown in the figure on side l, one rotates the CD rod counter-clockwise, the AB rod also begins to rotate counter-clockwise. However, at a certain point the latter inverts its direction of motion, although CD continues its counter-clockwise motion. It is not easy to imagine the movement of this simple mechanism - it's be best to make a simple model with cardboard strips or Meccano rods or some such, or to look at the figure on the side. The leg of the cyclist and the arm of the pedal form an articulated quadrilateral very similar to this one.

The difficulty in predicting the type of movement depends also on the fact that, by varying the rods' length ratio, the system may function in a very different way. This is the fascinating, interesting and useful nature of the articulated quadrilateral.

If the opposed rods are of equal length, the system becomes an articulated parallelogram, and the opposed rods always remain parallel. This is the basis of numerous applications of the mechanism which are under our eyes every day. Drawing boards, scales, Venetian blinds, the windscreen wipers of a bus (and of a cars, but in a less obvious way) - all use this simple mechanism, which is also found in some table lamps, in lifting machinery, in bicycles gearboxes, in sewing baskets...
In each case in which it is necessary that some parts keep a prefixed position on a plane, the articulate parallelogram offers the optimal solution. For example, in the case of the machinery used to lift personnel to check street lighting, it is necessary to ensure that the cabin stay horizontal, no matter the height from the ground. Please note that some of the described examples use two coupled parallelograms - in this way one may obtain movements in two directions (with two degrees of freedom) but the parallel nature of movement remains guaranteed.

The pantograph is itself practically based on a double articulated parallelogram (in the case of Figure 1, it is a rhombus). If PC = CD = DE = EB = AE = AC, then the points P, A, B are always on a straight line and the distance of B from the fixed point P is always twice that of A, so that, if A draws a curve, B draws a similar curve, augmented by a factor of 2.
The figure on the side shows the diagram of a pantograph in a configuration such that the augmentation factor is four.

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In the general case, as we have seen, the articulated quadrilateral works differently according to the ratios between the rods' lengths. This is not the time and place for a detailed analysis of the various types. We will simply try to show the great versatility of this mechanism. Often it is used as a force multiplier, in the cases when the movement is caused by a shaft rotating at near-constant speed, while the force must be applied only on a short distance with small movements, as it happens in oil-mills or in some cutting devices. Or, on the contrary, this mechanism may be used to transform alternating motion in circular motion, as in the pedals of old sewing machines, or with even more discreet functions, such as in rubbish bins.

Also of interest is the use of the articulated quadrilateral in the project for a car's suspension (see figure) such that the wheels may absorb all the shocks due to the road's irregularities, without directly transmitting them to the passengers. In this case, one must ensure that both wheels maintain the same vertical angle.

If one considers a joint plane with the rod CB, leaving AD fixed and rotating the shorter rod, the points of this plane draw very diverse curves, according to the position of the point. To individuate the point tracking the plane joint to the BC rod, it is sufficient to add to the mechanism other two rods BE and CE that form a triangle with BC.
The following figure gives an idea of this phenomenon: some curves are shaped like a stretched 8, other are egg-shaped, other have an almost rectilinear part.

The latter opportunity is exploited e.g. in the film pulling mechanism in motion picture projectors and cameras (figure). When B performs a full revolution around A, the point E draws the dotted trajectory, causing the sudden advancement of the film during its near-rectilinear part, then frees itself and moves back on the other half of the trajectory.

A crane's arms also takes advantage of this property. In this case the problem lies in ensuring that, varying the distance from the base, the height of the load remains basically unchanged.

Obviously, more complex mechanisms can perform more complex functions, but the problem there is not different in principle from the simple cases which we have analysed: what one wants to obtain from a mechanism is to have a given point draw a given trajectory which describes its function, without needing this trajectory to exist physically.

As we have seen, Kempe has demonstrated that there exists a connecting rod mechanism to trace any algebraic curve. However, in general such a mechanism is constituted by a very high number of rods. Having determined more and more simple and functional mechanisms to perform the same function, however approximately, has been one of the fundamentals of technological progress.