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projecting lines converge towards some fixed point, the
* This reference is to the Elements, article 208.
In the projection of geometrical magnitudes gener- Fig. 1.
ally, the planes of projection may be so chosen that the points to be projected shall be all in the first diedral angle. The projection consists of two parts. First, the several points of the body are projected by perpendiculars drawn from them to the horizontal plane ; the intersections of these perpendiculars with the horizontal plane, make their projection upon this plane : This is called their horizontal projection. Secondly, the several points are projected by perpendiculars drawn to the vertical plane : the intersections of these perpendiculars with the vertical plane, constitute the vertical projection of these points.
6. This projection is used by architects and engineers in representing the several parts of any edifice or structure. A drawing is made upon paper, which is a miniature representation of a horizontal projection of the edifice. This is called the ground plan, or simply the plan of the edifice ; and exhibits the relative situations of all the remarkable points in the edifice, referred to a horizontal plane by perpendiculars to this plane. The plan of an edifice, it will be perceived, shows us nothing respecting the height of its different parts. In order to fix the heights, and therefore to determine the relative positions of the several points in the edifice, as well as its various dimensions, a drawing is also made representing a vertical projection of these points. When the vertical projection represents the interior of the building, it is called a section or profile. When this projection exhibits the exterior of the building, it is called an elevation. It is manifest that those lines in the building which are oblique to each of the planes of projection, will not be projected in their true magnitudes upon either of the planes. We shall see, however, that descriptive geometry accurately determines these magnitudes notwithstanding this apparent defect of the process.
7. Let M (fig. 1) be a point in the first angle. Draw MM’ perpendicular to the horizonal plane, and MM" perpendicular to the vertical plane; M' is the horizontal projection of the point M, and MI" is the vertical projection of this point. If we suppose a plane to pass through
the projecting lines MM', MM", it will be perpendicular to each of the planes of projection BC, BD, (El. 207), and therefore perpendicular to the ground line (El. 209). But if MM" be perpendicular to AB, by revolving the vertical plane, about the ground line, through the second diedral angle into the plane of the paper, MM" will be in the same straight line with MM’ also perpendicular to AB (El. 14). We therefore say—The vertical and horizontal projections of the same point, are in a straight line perpendicular to the ground line.
8. Remark. The distance of the vertical projection of a point from the ground line is equal to the distance of the point itself from the horizontal plane ; and the distance of the horizontal projection of a point from the ground line, is equal to the distance of the point, in space, from the vertical plane.
It follows, therefore, that all points situated in the vertical plane, have their horizontal projections in the ground line ; and all points situated in the horizontal plane, have their vertical projections in the ground line. All points in either plane are their own projections in that plane.
9. The two projections of a point determine its position ; for M" and M" being the two projections of a point, if we draw through the projections straight lines perpendicular to each of these planes, these perpendiculars must pass through the proposed point; it must therefore be at their intersection M.
When we speak of a point in space given in position by its projections, we say, the point (M', M*); by which we mean, the point whose horizontal projection is M. and vertical projection M".
10. Two lines which are parallel or which cut each other, determine the position of a plane (El. 197). If, therefore, we have the intersections of any plane with the two planes of projection, this plane is given in position.
11. The two intersections of the proposed plane with the planes of projection, are called its traces upon these planes. Its intersection with the vertical plane is called its vertical trace; and its intersection with the horizontal plane, is its horizontal trace. If the proposed plane be parallel to one of the planes of projection it can
have no trace upon that plane. Therefore, if a plane Fig. 1. has but one trace, it is parallel to that plane of projection which contains no trace. A plane whose horizontal trace is M'N', and whose vertical trace M"N", is called usually, the plane (MN", M'N"). 12. If a plane be inclined to each of the planes of projection, but parallel to the ground line, it can never meet the ground line (El. 198); and consequently its traces upon the planes of projection cannot meet the ground line ; they are therefore parallel to it: And we say— A plane parallel to the ground line, will have its traces parallel to the ground line. And the traces of a plane inclined to this line, will intersect it. 13. If the two traces of the proposed plane are perpendicular to the ground line, the plane itself must be perpendicular to the ground line (El. 199); and if the ground line is perpendicular to the proposed plane, the co-ordinate planes, both passing through this line, must be perpendicular to the proposed plane (El. 207). Therefore —If the two traces of the proposed plane are perpendicular to the ground line, the plane itself is perpendicular to each of the co-ordinate planes.
14. As one point is projected by a perpendicular drawn to the plane of projection ; all the points in a straight line are projected by perpendiculars drawn to the plane of projection (fig. 2). These perpendiculars Fig. 2. being parallel, and passing through the same straight line, are in the same plane. We therefore say—The horizontal projection of a straight line, is the horizontal trace of a plane passing through the line, and perpendicular to the horizontal plane. The vertical projection of a straight line, is the vertical trace of a plane passing through the line, and perpendicular to the vertical plane. These planes are called projecting planes.*
15. It is here supposed that the proposed line is not perpendicular to either of the co-ordinate planes. The projection of a straight line perpendicular to the plane of projection, is a point.
16. If a line be parallel to either of the co-ordinate
* It is important to distinguish between projecting planes and the planes of projection.
Fig. 2. planes, its projection upon the other plane will be parallel to the ground line. For the line itself and the projecting line of any point in it, are two lines which determine this projecting plane to be parallel to one of the planes of projection (El. 197); their intersections with the other plane of projection are therefore parallel (El. 214). From this it follows that—The projections, on the same plane, of parallel lines, are parallel. If, however, the lines are perpendicular to either of the co-ordinate planes, their projections will be points, and cannot be said to be parallel.
17. Two projections of a line determine its position in space. For drawing a plane through its horizontal projection perpendicular to the horizontal plane, the plane must pass through the proposed line; and passing a plane through the vertical projection perpendicular to the vertical plane, this plane will also pass through the line; the line must therefore be the intersection of these two planes, which is determined, the planes themselves being determined : The line is therefore given in position by its projections. Remark. It is readily seen that each of the projections of the proposed line, answers to an infinite number of lines all in this projecting plane; but the two projections answer only to a line in each of the planes; this line must be their intersection. If the proposed line were in a plane perpendicular to each of the co-ordinate planes, and inclined to these planes, the two projections would not determine it. Some other condition would be necessary; such as its projection upon a third plane, or the distances of two points of the line from their projection on one of the planes. When we have occasion to speak of a line given by its horizontal projection M'N', and its vertical projection M"N", we say, the line (M'N', M"N"). 18. [Right lines and planes are considered as indefinite in extent. In the diagrams, those parts of the given or the required lines which are in the first angle, are made full ; the dotted lines in the figures represent portions of the proposed lines which are situated behind a plane, and supposed to be seen through the plane. Those