Plan, Elevation, Projections, Planes of Projection, Projectors.-The view of the table, and therefore of any solid, as seen from above is called its plan, and the view as seen from the end its elevation. The views of objects drawn on the principles of solid geometry are called its projections, the imaginary planes on which they are drawn are called the planes of projection, and the lines from the object to the planes are called projectors. We have seen that only two projections are necessary— -one on the floor, a horizontal plane, the other on the wall, a vertical plane, and as these two planes are always required for the plan and elevation of a solid, they are termed the horizontal plane of projection, usually denoted by the capital letters HP, and the vertical plane of projection, usually abbreviated to V P. Evidently the two planes of projection intersect in a line, which, owing to the horizontal plane being supposed the plane of the ground, is called the ground line, and is generally denoted by the capital letters XY.

All this will be clearly understood by reference to the following example.—In Fig. 60a is shown a representation of a simple

solid, an equal armed cross made of square wood, with the horizontal and vertical planes of projection, having its plan and elevation drawn upon them Lines perpendicular to the HP are

drawn through each corner of the solid, meeting the H P in the points marked a, b, c, d . . ., and similarly lines perpendicular to the V P are drawn through each corner, meeting the V P in the points marked a', b', c', d'. . . To join these points in the right, order we look at the solid, and see that A joins B, and that B joins C, and C joins D, and so on, and, therefore, by joining a to b, and a to b', &c., we obtain on the H P and V P a plan and an elevation of the solid. Notice that the plan really represents. two faces of the cross, the upper and lower, which are similar, and, therefore, that each point in the plan shows at least two corners of the cross, and similarly with the elevation.

On the right hand of the cross is shown a point P with its plan p, and its elevation p', the line Pp being its projector to the HP, and the line Pp' its projector to the V P. The plan and elevation of these projectors are drawn at po and p'o, and it should be specially noticed that they make two lines, meeting on the ground line, each being perpendicular to it. The student should suppose that the plan and elevation of the projectors of the cross are drawn in this way, although in the figure they are omitted for the sake of clearness.

Now, imagine the VP to be turned upon the ground line as a hinge, away from the solid, as shown by the arrow, until it becomes horizontal and forms a continuation of the HP. We shall then have the representation of Fig. 606, which is the usual solid geometry projection or drawing, showing a plan and elevation upon a single flat sheet of paper. With the aid of these two figures the student should now verify the following X statements, all of which are important and should be remembered:

(a) The plan is below the ground line, and the elevation above it.

(b) The plan and elevation of the same point are exactly one under the other, in a line perpen

dicular to the ground line, therefore the plan of a solid should be directly under the elevation.

(c) Heights above the HP are shown in the elevation.
(d) Distances in front of the VP are shown in the plan.

(e) The projectors are shown by lines, joining points in the plan and elevation, and perpendicular to the ground line.

(f) The elevation of a point in the H P, and the plan of a point in the V P, are shown by a point on the ground line, for if p and p' be these points, their elevation and plan are both shown by the point o (Fig. 606). From this it follows that when a solid has one face or edge in the ground plane or H P, its elevation will begin from the ground line, and similarly if it has a face or edge in the VP its plan will also begin from the ground line.

We see from Fig. 606 that the plan and elevation are separated from one another, and that the distance between them depends only on the height of the solid above the HP and its distance in front of the V P.

In examples of solid geometry these distances may be given of any desired length, or may be left to the will of the student, in which case it is convenient to assume the solid, as standing on the HP and in front of the V P, as this gives an elevation starting from the ground line and a plan removed from it, thus separating the two drawings and adding to their clearness.

Marking Plans and Elevations.-It will have been noticed in Fig. 60a that each point in the solid is denoted by a capital letter, as A, B, C, while its plan is marked by the same letter in small type, as a, b, c, and its elevation by a similar letter with the addition of a dash, as a', b', c'. This is a convenient notation, usually adopted in solid geometry, and will be adhered to in all following examples.

A solid is bounded by surfaces, a surface by lines, and a line by points, and we shall, therefore, lead up to the projection of solids by examples dealing with points, lines, and surfaces.

In commencing solid geometry it will be found very helpful to make up a rough model of the planes of projection, and of the objects to be drawn. A book or instrument box opened at right angles very well represents the HP and V P, a drawing pin may represent a point, a pencil a line, and a set square or piece of card a surface or plane, while models of simple solids can be easily made. It is only in this way that the beginner can hope to gain an intelligent and useful knowledge of the subject, and be able to proceed with confidence to advanced problems and to machine drawing, where the objects to be drawn exist only as a mental picture, and where their positions relative to, and their projections upon, the planes of projection have to be vividly imagined before they can be represented upon the paper. All engineering draughtsmen use the results of the principles of solid geometry, although, as the student will see in

due course, they appear to dispense with the actual use of projectors, ground line, planes, &c.

It will be seen from Fig. 61a that the HP and VP are carried on so as to extend on both sides of the ground line X Y. This is evidently correct, as a plane has no limit of either length or breadth. When thus regarded, the planes of projection are said to form four dihedral angles (angles formed by surfaces), and a point may be regarded as being in either one of the angles; as, for example, a point may be below the HP and behind the V P, and the position of its plan and elevation relative to the ground line are affected accordingly; but as this is a matter of theoretical rather than of practical importance, it will not be further considered, and reference will be made to the first dihedral angle only.

As the position of points, lines, and solids can only be stated as distances from the planes of projection, which, as we have seen, resolves itself into distances below and above the ground line, it is evident that in all examples we must commence by drawing the ground line. It should be noticed that when this is done the paper above the X Y represents the V P, and the paper below it the H P, and that if the paper be bent about the X Y, as a hinge, bringing the V P into a vertical plane, it will represent a model of the planes of projection.

The following points should be particularly observed:All construction lines, such as projectors, should be drawn as fine light lines, and the projections or plans and elevations of the line, figure, or solid being drawn, should be shown by dark lines.

Lines to represent the edges of a solid, not actually seen, owing to some part of the solid being between them and the eye of the observer, should be shown by dark dotted lines.

Projection of Points. To show the projections of a point given its distance above the HP and in front of the V P, first draw the XY, then through any point in it draw a perpendicular line to represent the projectors of the point, mark a point in this line above the X Y, equal to the height of the point above the H P, and a point in the line below the X Y, equal to the distance of the point in front of the V P. If the point is denoted by the letter P, its plan should be marked p and its elevation p'. When the distances are given in the question, it is better that they should be marked on the drawing, using dimension lines with arrow heads as in Fig. 50. The student should also aim at writing above the drawing a brief description of what the drawing represents (not a mere copy of the question), whether of a point, line, surface, or solid, and its special position

relative to the planes of projection, as it is just as important to know exactly what position is indicated by a given drawing as to be able to make the drawing of a solid in a given position.

EXAMPLES.

EX. 1.-Draw the projections of the following points distinguishing the plan and elevation of each :

(a) Point A in both planes.

(b) Point B in H P, 13" in front of V P. (c) Point C in V P, 13′′ above H P.

(d) Point D, 18" from both planes.

(c) Point E, 2·6′′ above H P, 1·7′′ in front of V P.
(f) Point F, 2.4" below H P, 1.5" behind V P.
(g) Point G, 2′′ above H P, 1.9" behind V P.

(h) Point H, 21" below H P, 1.6" in front of V P.

Projection of Lines.-Lines may be parallel to, perpendicular to, or inclined to either the HP or V P, and in some cases to both. Lines may also be contained by, or may lie in, either

or both of the planes. As the ends of a line are points, a line is spoken of as the line A B or CD, one letter being marked at each end, its plan is then marked ab or cd, and its elevation a'b' or c'd'. A line is fixed by stating its position relative to