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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.

Point B in H P, 18" in front of V P.

Point C in V P, 13′′ above H P.

(d) Point D, 1.8′′ from both planes.

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. (9) Point G, 2" above H P, 1.9′′ behind V P.

Point H, 21" below HP, 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

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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

the planes of projection, both with regard to its inclination to them, and its distance from them, hence these conditions must be known before the projections of the line can be drawn.

Three lines, A B, CD, E F, each differently placed with regard to the planes of projection, together with their projectors, and their plans and elevations upon the HP and VP are represented in Fig. 61a in such a way as to show the principle of projection. In Fig. 616 the two planes are shown with the VP thrown down, thus forming one hori

zontal sheet, and showing the plans and elevations of the lines exactly as they should appear when drawn upon the paper. The positions of the lines are as follows:—

A B is perpendicular to the HP and above it, parallel to the V P and in front of it.

CD is parallel to the

Fig. 616.

HP and above it, inclined to the V P and in front of it.
EF is inclined to both planes, and removed from both.

With the help of these figures and of a model of the planes and a pencil to represent a line, the student should carefully verify the following statements:

(a) When a line is parallel to, or is contained by, the HP its plan is equal in length to the line itself.

(b) When a line is parallel to, or is contained by, the VP its elevation is equal in length to the line itself.

(c) When a line is parallel to both planes its plan and elevation are equal in length to the line itself.

Therefore, when a line is parallel to, or is contained by, a plane, its projection upon that plane is a line equal in length to the line itself.

(d) When a line is inclined to the HP its plan is shorter than the line itself.

(e) When a line is inclined to the VP its elevation is shorter than the line itself.

(f) When a line is inclined to both the HP and V P its plan and elevation are both shorter than the line itself.

Therefore, when a line is inclined to a plane its projection upon that plane is a line, the length of which is less than the length of the line itself.

(g) When a line is perpendicular to the HP its plan is a point.

(h) When a line is perpendicular to the VP its elevation is a point.

Therefore, when a line is perpendicular to a plane its projection upon that plane is a point.

(i) When a line is contained by both the HP and V P its plan and elevation coincide in the ground line.

(j) When a line is inclined to the HP and parallel to the V P its inclination is shown in the elevation.

(k) When a line is inclined to the VP and parallel to the HP its inclination is shown in the plan.

Therefore, when a line is inclined to one of the planes of projection and parallel to the other, its inclination is shown upon the plane to which it is parallel. It will be seen later that when a line is inclined to both planes its inclination is not shown either in the plan or elevation.

The projections of a line should present no difficulty if it is remembered that the ends of the lines are points, whose projections can be found as already described. For if the plan and elevation of the points be joined, the joining lines will be the plan and elevation of the line having the points for its ends. Notice also, that when a line is inclined to one of the planes, its projection upon the other plane must be drawn first.

EXAMPLES.

EX. 2.-Draw the projections of a line 31" long, in the following positions, mark each end of the line in plan and elevation with letters, and mark the lengths and inclination of the lines. Write above each its position with regard to the HP and V P. (a) Parallel to both planes and in both.

(b) Parallel to both planes and 1.6′′ from each.

(c) Parallel to both planes, 1" above H P, 2.3′′ in front of V P. (d) Parallel to both planes, 2′′ above H P, 1·7′′ in front of V P. (e) Inclined 60° to H P, one end in HP; parallel to V P, 1.3′′ in front.

(f) Inclined 45° to HP, one end 1.4" above HP; parallel to V P, 11" in front.

(g) Inclined 60° to V P, one end in V P; parallel to H P, 1.3′′

above.

(h) Inclined 45° to V P, one end 1" in front of V P; parallel

to HP and in H P.

(i) Parallel to V P and 1" in front, its ends 1" and 21′′ above HP. Show angle of inclination to H P.

(j) Parallel to HP and 11" above, its ends 1.3′′ and 2·7′′ in front of V P. Show angle of inclination to V P.

(k) Perpendicular to H P and in V P, one end in H P. (7) Perpendicular to H P, one end 11′′ above, 1′′ in front of V P. (m) Perpendicular to V P, one end in V P, 11′′ above H P. (n) Perpendicular to V P and in HP, one end 1" in front of V P.

EX. 3. The projectors of a line are 2" apart, measuring along the XY. The line is parallel to the V P, and 12" in front, and is inclined at 60° to the HP, one end being in the VP. Draw the plan and elevation.

Traces of Lines.—When a line is inclined to a plane, it will evidently meet that plane if produced far enough. The point where the line meets the plane is called its trace. The horizontal trace, H T, of a line, is the intersection of the line with the H P, and the vertical trace, V T, its intersection with the V P.

If this definition is understood, no difficulty should be experienced in finding the traces of a line. When a line is parallel to a plane, it will, of course, have no trace upon that plane, but when it is inclined to a plane with one end in the plane, that point is its trace upon that plane, therefore, when a line is inclined to a plane without meeting it, it has only to be produced to meet the plane, and its trace will be the meeting point. Notice that the HT of a line must be in its plan, or the plan produced, and the V T of a line must be in its elevation, or the elevation produced. Thus, in Fig. 626, the traces of the doubly inclined line A B are at the points marked HT and VT, the manner in which these traces are found being clear from the construction.

EXAMPLES.

EX. 4.-Draw the projections of a line as in Ex. 2, e, f, k, l, and find the H T of the line.

EX. 5.-Draw the projections of a line as in Ex. 2, g, h, m, n, and find the V T of the line.

EX. 6. The end A of a line AB is 1" above the HP and 21" in front of the V P, the end B is 23" above the HP and " in front of V P, the projectors measuring along the X Y being 21" apart. Draw the projections of the line, and find its H and V T.

True lengths of Lines.-We have seen that when a line is inclined to both the HP and V P, its true length and inclinations are not seen, either in the plan or elevation, and we must now examine a method whereby the true length and inclination of the line can be ascertained. As there are conditions which occur in other problems of practical importance besides those of

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