Authorities.—In this article we have given a purely geometrical theory of conics, cones of the second order, quadric surfaces, &c. In doing so we have followed, to a great extent, Reye’s Geometrie der Lage, and to this excellent work those readers are referred who wish for a more exhaustive treatment of the subject. Other works especially valuable as showing the development of the subject are: Monge, Géométrie descriptive: Carnot, Géométrie de position (1803), containing a theory of transversals; Poncelet’s great work Traité des propriétés projectives des figures (1822); Möbins, Barycentrischer Calcul (1826); Steiner, Abhängigkeit geometrischer Gestalten (1832), containing the first full discussion of the projective relations between rows, pencils, &c.; Von Staudt, Geometrie der Lage (1847) and Beiträge zur Geometrie der Lage (1856–1860), in which a system of geometry is built up from the beginning without any reference to number, so that ultimately a number itself gets a geometrical definition, and in which imaginary elements are systematically introduced into pure geometry; Chasles, Aperçu historique (1837), in which the author gives a brilliant account of the progress of modern geometrical methods, pointing out the advantages of the different purely geometrical methods as compared with the analytical ones, but without taking as much account of the German as of the French authors; Id., Rapport sur les progrès de la géométrie (1870), a continuation of the Aperçu; Id., Traité de géométrie supérieure (1852); Cremona, Introduzione ad una teoria geometrica delle curve piane (1862) and its continuation Preliminari di una teoria geometrica delle superficie (German translations by Curtze). As more elementary books, we mention: Cremona, Elements of Projective Geometry, translated from the Italian by C. Leudesdorf (2nd ed., 1894); J. W. Russell, Pure Geometry (2nd ed., 1905). (O. H.)
III. Descriptive Geometry
This branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane. The most important method is that which was invented by Monge towards the end of the 18th century. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a projection is called orthographic (see Projection, § 18). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge’s method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection.
§ 1. We suppose then that we have two planes, one horizontal,
the other vertical, and these we call the planes of plan and of elevation
respectively, or the horizontal and the vertical plane, and
denote them by the letters π1 and π2. Their line of intersection is
called the axis, and will be denoted by xy.
If the surface of the drawing paper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis xy. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is called rabatting one to the other. Of course there is no necessity to have one of the two planes horizontal, but even when this is not the case it is convenient to retain the above names.
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| Fig. 37. | Fig. 38. |
The whole arrangement will be better understood by referring to fig. 37. A point A in space is there projected by the perpendicular AA1 and AA2 to the planes π1 and π2 so that A1 and A2 are the horizontal and vertical projections of A.
If we remember that a line is perpendicular to a plane that is perpendicular to every line in the plane if only it is perpendicular to any two intersecting lines in the plane, we see that the axis which is perpendicular both to AA1 and to AA2 is also perpendicular to A1A0 and to A2A0 because these four lines are all in the same plane. Hence, if the plane π2 be turned about the axis till it coincides with the plane π1, then A2A0 will be the continuation of A1A0. This position of the planes is represented in fig. 38, in which the line A1A2 is perpendicular to the axis x.
Conversely any two points A1, A2 in a line perpendicular to the axis will be the projections of some point in space when the plane π2 is turned about the axis till it is perpendicular to the plane π1, because in this position the two perpendiculars to the planes π1 and π2 through the points A1 and A2 will be in a plane and therefore meet at some point A.
Representation of Points.—We have thus the following method of representing in a single plane the position of points in space:—we take in the plane a line xy as the axis, and then any pair of points A1, A2 in the plane on a line perpendicular to the axis represent a point A in space. If the line A1A2 cuts the axis at A0, and if at A1 a perpendicular be erected to the plane, then the point A will be in it at a height A1A = A0A2 above the plane. This gives the position of the point A relative to the plane π1. In the same way, if in a perpendicular to π2 through A2 a point A be taken such that A2A = A0A1, then this will give the point A relative to the plane π2.
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| Fig. 39. |
§ 2. The two planes π1, π2 in their original position divide space into four parts. These are called the four quadrants. We suppose that the plane π2 is turned as indicated in fig. 37, so that the point P comes to Q and R to S, then the quadrant in which the point A lies is called the first, and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane. Now we go round the axis in the sense in which the plane π2 is turned and come in succession to the second, third and fourth quadrant. In the second a point lies above the plane of the plan and behind the plane of elevation, and so on. In fig. 39, which represents a side view of the planes in fig. 37 the quadrants are marked, and in each a point with its projection is taken. Fig. 38 shows how these are represented when the plane π2 is turned down. We see that
A point lies in the first quadrant if the plan lies below, the elevation above the axis; in the second if plan and elevation both lie above; in the third if the plan lies above, the elevation below; in the fourth if plan and elevation both lie below the axis.
If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself. If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it.
Of each of these propositions, which will easily be seen to be true, the converse holds also.
§ 3. Representation of a Plane.—As we are thus enabled to represent points in a plane, we can represent any finite figure by representing its separate points. It is, however, not possible to represent a plane in this way, for the projections of its points completely cover the planes π1 and π2, and no plane would appear different from any other. But any plane α cuts each of the planes π1, π2 in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane α.
A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines which meet on the axis determine a plane.
If the plane is parallel to the axis its traces are parallel to the axis. Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. Thus a plane parallel to the horizontal plane of the plan has only one finite trace, viz. that with the plane of elevation.
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| Fig. 40. |
If the plane passes through the axis both its traces coincide with the axis. This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third plane and denote it by π3. As it is perpendicular to π1, it may be taken as the plane of elevation, its line of intersection γ with π1 being the axis, and be turned down to coincide with π1. This is represented in fig. 40. OC is the axis xy whilst OA and OB are the traces of the third plane. They lie in one line γ. The plane is rabatted about γ to the horizontal plane. A plane α through the axis xy will then show in it a trace α3. In fig. 40 the lines OC and OP will thus be the traces of a plane through the axis xy, which makes an angle POQ with the horizontal plane.
We can also find the trace which any other plane makes with π3. In rabatting the plane π3 its trace OB with the plane π2 will come to the position OD. Hence a plane β having the traces CA and CB will have with the third plane the trace β3, or AD if OD = OB.
