Let A1 A2 (fig. 42) be the given point, α′ α″ the given plane, a line l1 through A1, parallel to α′ and a horizontal line l2 through A2 will be the projections of a line l through A parallel to the plane, because the horizontal plane through this line will cut the plane α in a line c which has its horizontal projection c1 parallel to α′.
§ 12. We now come to the metrical properties of figures.
A line is perpendicular to a plane if the projections of the line are perpendicular to the traces of the plane. We prove it for the horizontal projection. If a line p is perpendicular to a plane α, every plane through p is perpendicular to α; hence also the vertical plane which projects the line p to p1. As this plane is perpendicular both to the horizontal plane and to the plane α, it is also perpendicular to their intersection—that is, to the horizontal trace of α. It follows that every line in this projecting plane, therefore also p1, the plan of p, is perpendicular to the horizontal trace of α.
To draw a plane through a given point A perpendicular to a given line p, we first draw through some point O in the axis lines γ′, γ″ perpendicular respectively to the projections p1 and p2 of the given line. These will be the traces of a plane γ which is perpendicular to the given line. We next draw through the given point A a plane parallel to the plane γ; this will be the plane required.
Other metrical properties depend on the determination of the real size or shape of a figure.
In general the projection of a figure differs both in size and shape from the figure itself. But figures in a plane parallel to a plane of projection will be identical with their projections, and will thus be given in their true dimensions. In other cases there is the problem, constantly recurring, either to find the true shape and size of a plane figure when plan and elevation are given, or, conversely, to find the latter from the known true shape of the figure itself. To do this, the plane is turned about one of its traces till it is laid down into that plane of projection to which the trace belongs. This is technically called rabatting the plane respectively into the plane of the plan or the elevation. As there is no difference in the treatment of the two cases, we shall consider only the case of rabatting a plane α into the plane of the plan. The plan of the figure is a parallel (orthographic) projection of the figure itself. The results of parallel projection (see [Projection], §§ 17 and 18) may therefore now be used. The trace α′ will hereby take the place of what formerly was called the axis of projection. Hence we see that corresponding points in the plan and in the rabatted plane are joined by lines which are perpendicular to the trace α′ and that corresponding lines meet on this trace. We also see that the correspondence is completely determined if we know for one point or one line in the plan the corresponding point or line in the rabatted plane.
Before, however, we treat of this we consider some special cases.
§ 13. To determine the distance between two points A, B given by their projections A1, B1 and A2, B2, or, in other words, to determine the true length of a line the plan and elevation of which are given.
| Fig. 43. |
| Fig. 44. |