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.
| 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.