CHAPTER XI

PROJECTION

134. Descriptive Geometry.-Practical Solid Geometry, or Descriptive Geometry is that branch of geometry which treats (1) of the representation, on a plane surface, of points, lines, and figures in space, in such a way that the relative positions of the points, lines, and figures, and also the exact forms of the lines and figures are determined, and (2) of the graphic solution of problems connected with points, lines, and figures in space. The problems of descriptive geometry are best solved by means of the method of projections.

135. Projection.-When an object is seen by the eye of a spectator, rays of light come from all the visible parts of the object and converge towards a point within the eye. Now suppose that a

flat sheet of glass is placed between the object and the eye of the spectator, and that each ray of light, in passing through the glass from the object to the eye, leaves a mark on the glass of the same colour and tint as the part of the object from which the ray came. In this way a picture would be produced on the surface of the glass, and if the object be removed while the picture and the eye remain stationary, the picture would convey to the mind of the spectator the same knowledge of the object as was conveyed by the presence of the object itself. Again, if instead of the rays of light from all the visible points of the object leaving an impression on the glass, only those which came from the edges of the object were to do so, an outline would be produced on the surface of the glass which, although it would not convey to the mind of the spectator the same impression as the presence of the object itself might still give a good idea of its form.

The foregoing remarks are illustrated by Fig. 286, where AB represents an object viewed by an eye at E; CD is a plane interposed between E and AB; the thin dotted lines represent a few of the rays of light passing from the edges of the object to the eye, and A'B' is the outline obtained from the intersections of the rays of light with the plane CD. The figure A'B' is called a projection of the object AB on the plane CD.

The plane upon which a projection is drawn is called a plane of projection.

The rays of light or imaginary lines passing from the different points of the object to the corresponding points of the projection are called projectors.

When the projectors converge to a point the projection is called a radial, conical, or perspective projection.

When the point to which the projectors converge is at an infinite distance from the object the projectors become parallel, and the projection is called a parallel projection.

If besides being parallel the projectors are also perpendicular to the plane of projection the projection becomes a perpendicular, an orthogonal, or an orthographic projection.

For the purposes of descriptive geometry orthographic projections are the most convenient and most commonly used, and when the term projection is used without any qualification orthographic projection is generally understood. In what follows projection will mean orthographic projection.

The projection of a point upon a plane is the foot of the perpendicular let fall from the point on to the plane.

The projection of a line upon a plane is the line which contains the projections of all the points of the original line.

The projecting surface of a line is the surface which contains the

projectors of all the points of that line. When the projecting surface of a line is a plane it is called the projecting plane of the line. The projecting surface of a straight line is always a plane, but a line is not necessarily straight because its projecting surface is a plane. These definitions of projecting surface and projecting plane of a line and the statements which follow them apply to all kinds of projection.

One projection alone of a figure is not sufficient for determining its exact form. For example if a triangle abc drawn on a sheet of paper be taken as the projection on the paper of a triangle ABC somewhere above it, it is clear that the exact form of the triangle ABC will depend on the relative distances of its angular points from the paper, but the projection abc gives no information about these distances. If, however, another projection a'b'c' of the triangle ABC be obtained on a sheet of paper at right angles to the former one, then, as will be shown later, the true form of the triangle ABC may be obtained from these two projections.

The representation of an object by means of two projections, one on each of two planes at right angles to one another, and how these

projections are drawn on a flat sheet of paper will be understood by reference to Figs. 287 and 288.

Fig. 287 is to be taken as a pictorial projection of a model. Two planes of projection are shown one being vertical and the other horizontal. These planes, called co-ordinate planes, divide the space surrounding them into four dihedral angles or quadrants which are named, first, second, third, and fourth dihedral angles or first, second, third, and fourth quadrants. If an observer be facing the vertical plane of projection, then the first quadrant is above the horizontal plane of projection and in front of the vertical plane of projection. The second quadrant is behind the first and the others follow in order

as shown. The line of intersection of the planes of projection is called the ground line and is lettered XY.

An object is shown in the first quadrant and projections of it on the horizontal and vertical planes of projection are also shown. The projection on the horizontal plane is called a plan and the projection on the vertical plane is called an elevation.

Now imagine the vertical plane to turn about XY as an axis, carrying with it the elevation, until it is in a horizontal position. The horizontal and vertical planes of projection will now coincide and the plan and elevation of the object will be on one flat surface and their exact forms may be drawn as shown in Fig. 288.

Instead of imagining the vertical plane to turn about XY until it is horizontal, the horizontal plane may be imagined to turn about XY until it is vertical as shown in Fig. 289.

In British and European countries the general practice in making working drawings is to conceive the object to be placed in the first quadrant as shown in Fig. 287 and the working plan and elevation are then in the positions shown in Fig. 288, the plan being below and the elevation above XY. In the United States of America the practice of conceiving the object to be placed in the third quadrant as shown in Fig. 289 is now very general and the working plan and elevation are then in the positions shown in Fig. 290, the plan being above and the elevation below XY.

Whether the object be placed in the first quadrant or in the third quadrant it is supposed to be viewed from above in obtaining the plan, consequently when the object is in the first quadrant it lies between

the observer and the plane of projection and the projectors from the visible parts have to go through the object to the plan, but when the object is in the third quadrant the plane of projection lies between the observer and the object, hence the projectors from the visible parts to the plan are not obstructed by the object. In like manner for the elevation, the projectors from the visible parts of the object go through the latter when it is in the first quadrant but are clear of it when in the third quadrant; hence the advantage claimed for placing the object in the third quadrant. The practice of placing the object in the first quadrant is however so well established and the advantage claimed for placing it in the third quadrant is so small, being probably more imaginary than real, that it is doubtful whether a change from the older practice should be encouraged. In any case drawings on

either system can be made with equal facility if the principles are understood.

In working problems in descriptive geometry by the method of projections on co-ordinate planes, the given points and lines may be in any one quadrant, but the lines for the solution may extend into any or all of the other quadrants.

136. Notation in Projection. For the purpose of reference and for clearness, points, lines, and figures may be lettered. In general a point in space is denoted by a capital letter, its plan by a small italic letter, and its elevation by a small italic letter with a dash over it. Thus P denotes a point in space, p its plan and p' its elevation. A line AB in space would have its plan lettered ab and its elevation a'b'. A point P in space may be referred to as the point P or as the point pp'. In like manner a line AB in space may be referred to as the line AB or as the line ab, a'b'.

The horizontal and vertical planes of projection may be referred to by using the abbreviations H.P. and V.P. respectively.