11. The convention which serves as the basis of the method of projections is fit to express the position of a point in space, to express that of a straight line of finite or indefinite length and, in consequence, to represent the form and position of a body bounded by plane faces, by rectlineal edges, and by apexes of solid angles because, in this case, the body is entirely known when one knows the position of all its edges and all its angles. But if the body were bounded by a curved surface of which all the points were subjected to the same law, as in the case of a sphere, or by a discontinuous assemblage of parts with different curved surfaces, as in the case of a body fashioned on a lathe, this convention will not only be inconvenient, impractical and disadvantageous, but will be incapable of further development.
In the first place, it is easy to see that the convention which we have adopted will be inconvenient and impractical taken on its own, for to express the positions of all the points of a curved surface it is not only necessary that each of them shall be indicated by its horizontal and vertical projections, but that the two projections of the same point shall be joined to one another, in order that one will not confuse the horizontal projection of one point with the vertical projection of another; and the simplest means of connecting these two projections being to join them by a line perpendicular to the line of intersection of the two planes of projection, one will crowd drawings with a prodigious number of lines which will cause confusion, the greater as one approaches greater exactness. We shall see below that this method is insufficient and lacking in development potential.
Amongst the infinite number of different curved surfaces, there exist some which only extend to a finite part of space, and whose projections have a limited extent in all directions; the sphere, for example, is in this class. The extent of its projection onto a plane is reduced to a circle of the same radius as the sphere, and one can ensure that the plane upon which one must make the projection has dimensions large enough to take it. But all the cylindrical surfaces are indefinite in one direction – that of the line which serves as their generator. The plane itself, the simplest of surfaces, is unlimited in both directions. Lastly there exist a large number of surfaces which extend into all regions of space. But the planes upon which one executes projections are necessarily of limited extent. If one has no other means of making known the nature of a curved surface but the two projections of all the points, this method will only be applicable to those parts of the surfaces which correspond to the extent of the projections; all those points which lie beyond can never be expressed or known; the method is, therefore, insufficient. Lastly, it lacks elegance, because one can deduce nothing about planes tangent to the surfaces, about their normals, about the curvatures at each point, or indeed about any of the conditions it is necessary to consider when one wishes to investigate a curved surface.
It is necessary therefore to have recourse to a new convention which is compatible with the first, but which can supply that in which it is deficient. It is this new convention which we shall now introduce.
12. Every curved surface can be regarded as generated by the movement of a curved line, either constant in form while its position changes, or variable in form and position in space at the same time. As this proposition may be difficult to understand, because of its generality, we are going to explain it by taking a few examples with which we are already familiar.
Cylindrical surfaces may be generated in two principal ways; by the movement of a straight line which always remains parallel to a given straight line while it moves so as to pass through all the points of a given curve, or by the movement of the curve which in the first case served as a guide for the straight line, in such away that, one of its points moving along a given straight line, all the other points may describe lines parallel to this line. In either of these methods the generating line, which is a straight line in the first case and a curve in the second, is constant in form; it is constant in form and only changes its position in space.
Conical surfaces have the same two principal generations.
One can firstly regard them as produced by a line of indefinite length which, being subject to passing always through a given point, is placed such that it supports itself constantly on a given curve which guides it in its movement. The unique point through which the line passes is the centre of the surface; it is improperly that which is called the apex. In this generation, the generating line is again constant in form; it is always a straight line.
One can also produce conical surfaces in another way which, for greater simplicity, we will apply here only to the case of those which have circular bases. These surfaces can be regarded as the paths of the circumference of a circle placed with its plane always parallel to itself and with its centre always on the line running to the apex, its radiuses in each instant of movement being proportional to the distance of its centre from the apex. One sees that if, in its movement, the plane of the circle tends to approach the apex, the circle radius decreases to nothing when the plane passes through the apex, and that this radius changes its sense to increase indefinitely when the plane, having passed the apex, continues further. In this second generation, not only the circle circumference, which is the surface generator, changes position, but it changes its form at each instant of movement, since it changes its radius and, consequently, its curvature.
Let us quote a third example.
A surface of revolution can be produced by the movement of a plane curve which moves about a straight line positioned somewhere in the plane. Considering this method, the generating curve is constant in form and is only variable in position. But one can also regard it as produced by the circumference of a circle with its centre always on the axis, and its plane being always perpendicular to that axis, its radius being at each instant equal to the point where the plane of the circle cuts the axis and that where it cuts some given curve in space. Then the generating curve changes both in form and position.
These three examples must suffice to make clear that all curved surfaces can be produced by the movement of certain curved lines, and that there are none of which the form and position cannot be entirely determined by the exact and complete definition of its generation. It is this new conception which forms the complement to the method of projections. We will often have occasion later to assure ourselves of its simplicity and its potential.
It is not therefore, by giving the projections of individual points on a curved surface that one determines its form and position, but by being able to construct the generating curve through any point according to the form and position which belong to it in passing through this point. Here it should be observed, first, that each curved surface can be produced in an infinite number of ways and that the ingenuity and skill of the constructor are shown in choosing that which uses the simplest curve and requires the least trouble; second, that long usage has shown that in place of considering each curved surface as generated in one way only, which requires a study of the law of movement and a study of the change of form of the generator, it is often simpler to consider simultaneously two different generations, and to indicate for each point the constructions of two curved generators.
Thus, in descriptive geometry, to express the form and position of a curved surface it suffices, for any point on this surface and of which one of the porojections can be taken at plaeasure, to give the method of construction of the horizontal and vertical projections of two different generators which pass through this point.
13. Let us actually apply the above generalities to the plane, which is the simplest of surfaces and the one most frequently employed.
The plane is produced by a primary line whose first position is given, and which moves such that all its points describe lines parallel to a second given straight line. If the second line is itself in the plane considered, one can say also that this plane is produced by the second line, which moves such that all its points describe straight lines parallel to the first line.
One has, therefore, and idea of the position of a plane through the consideration of two straight lines, both of which can be regarded as the generator. The position of these two lines in the plane they produce is of no consequence. It is only necessary, then, in the method of projections, to choose those which require the simplest construction. And, for this reason, in descriptive geometry, one indicates the position of a plane by giving the two straight lines in which the plane cuts the planes of projection. It is easy to see that the two lines must meet at the same point on the intersection of the two planes of projection and that, consequentially, this point is where they must meet each other.
As we will very frequently have planes to consider, for shortness we will give the name traces to the lines where the plane cuts the planes of projection and which serve to indicate its position.