Monge's Procedure


Gaspard Monge, comte de Péluse (1746-1818) was a mathematician in the golden age of French mathematics, and a founder of the Ecole Polytechnique, the first modern engineering college. He is known for the science called Descriptive Geometry, which is a method of representing objects in a two-dimensional drawing in such a way that all of their geometric properties can be investigated and described. These methods were taught in classic form in every engineering course of study until computer graphics changed the approach. The fundamentals are the basis of computer graphics and design, however. The classic methods were intended as much as training for the mind as means of calculation. In addition, they are inherently fascinating and enjoyable.

The first attempt to represent a three-dimensional object in a two-dimensional drawing is by means of some pictorial view, such as perspective, isometric, or oblique views. In any such attempt, some information is necessarily lost, and the object is only partially or ambiguously represented. Antiquity used graphical methods almost exclusively in engineering calculations, a fact that was almost unrecognized, because of the perishability of such representations, and that the work was for practical, not literary, ends. Here and there, survivals have given a glimpse into the extensive development of these methods, in architecture and shipbuilding. Among these methods was the use of two coordinated drawings to present three-dimensional information, as mentioned by Vitruvius.

Monge revived this science, extending it to the solution of many types of problems, and showing how it gives a complete solution of the problem. We will show a simple application here that will give an idea of how it is carried out. In the Figure, AB is a plan view of, say, a mine shaft. The x and y coordinates of points A and B are represented in this view. Line 1-1 is called a folding line that separates the view showing the same points as A' and B'. This is a side view showing the coordinates y and z of the same two points. Note that the horizontal projection lines connecting A and A', and B and B', are perpendicular to the folding lines and show that the y-coordinates are the same. These two views give all three coordinates of any point in an object, and describe it completely. Of course, any two such views will suffice; we have made an arbitrary choice here. Suppose we want to know the length and slope of the tunnel between A and B. To find them graphically, we draw a folding line 2-2 parallel to the plan vies of the line. The direction perpendicular to this line must represent the z-coordinate. Now we draw the two projection lines AA" and BB", and measure out from folding line 2-2 exactly the same distances from folding line 1-1 to A' and B'. We have now located the projections of A and B in this view. The shaft lies in a plane parallel to this plane, so the true length and the true slope of the shaft are represented as shown, and can be scaled off.

We have shown the essential parts of Monge's Procedure, which are the selection of folding lines, and the measurement of equal distances in alternate views, by which any questions of size, shape, or orientation can be solved. This is the basis of orthogonal projection for engineering drawings, where three views are generally used to represent an object to make use of the redundancy for added vividness of presentation.


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Composed by J. B. Calvert
Created 19 June 1999
Last revised 20 February 2003