CHAPTER XIV

CHANGING THE PLANES OF PROJECTION

156. Auxiliary Projections.-In general the true form of an object is determined by a plan and an elevation, but there are cases where two projections are not sufficient, and in many other cases additional projections would make the form of the object much easier to understand. For example, in Fig. 354 are given a plan (a) and elevation (a) of a rectangular block having recesses in three of its faces. It is clear from the plan (a) and elevation (a') that the recess

in the top is rectangular and that the recess in the front is cylindrical, but the recess in the left hand end might be either rectangular or cylindrical. The form of the end recess is however determined by an end view (a) which is a projection of the block on a plane parallel to the ends. This second vertical plane of projection intersects the horizontal plane in a line X,Y, which is a second ground line. The relative positions of the solid and the three planes of projection are clearly shown in the pictorial projection, Fig. 355, where A.E.P. is the additional vertical plane of projection and may be called the auxiliary elevation plane.

157. Auxiliary Projections of a Point.-Referring to the pictorial projection, Fig. 356, A is a point in space, a is its projection

on the horizontal plane H.P. and a' is its projection on the vertical plane V.P. XY is the ground line in which these two planes of projection intersect. X,Y, is another ground line and is the line of intersection of the vertical plane A.E.P. with the horizontal plane H.P. a' is the projection of the point A on the plane A.E.P. If these two vertical planes of projection with the projections on them be rabatted about their respective ground lines into the horizontal plane, Fig. 357 is obtained. An examination of Figs. 356 and 357 will show that the distance of the auxiliary elevation a,' from X,Y, is equal to the distance of the elevation a' from XY; and just as the straight line joining a and a' is perpendicular to XY so also is the straight line joining a and a perpendicular to X,Y1.

Referring next to Figs. 358 and 359, A, a, and a' and the planes of projection H.P. and V.P. are the same as before, but X,Y, is now the line of intersection of a plane A.P.P. and the vertical plane of projection V.P., the plane A.P.P. being perpendicular to the plane V.P. When the planes H.P. and A.P.P. (Fig. 358) with the projections on them are rabatted about XY and X,Y, respectively into the vertical plane V.P., Fig. 359 is obtained. An examination of Figs. 358 and 359 will show that the auxiliary plan a, is at a distance from X,Y, equal to the distance of the plan a from XY, and just as the straight line aa' is at right angles to XY so also is the straight line a'a, at right angles to X,Y1.

It will be observed that the auxiliary plane of projection A.P.P. is not a horizontal plane, but it is usual to speak of the projection a, of the point A on this plane as an auxiliary plan.

The student must carefully study Figs. 356-359 and satisfy himself as to the correctness of the following rules which are used in the determination of auxiliary projections:

(1) The plan and elevation of a point are in a straight line which is perpendicular to the ground line.

(2) When a number of elevations are projected from the same plan, the distances of all the elevations of the same point from their corresponding ground lines are the same.

(3) When a number of plans are projected from the same elevation, the distances of all the plans of the same point from their corresponding ground lines are the same.

X

158. Distance of a Point from the Ground Line.-The distance of a point aa' (Fig. 360) from the ground line XY is the length of the perpendicular from the point on to XY. As in general this perpendicular will be inclined to both planes of projection, neither its plan nor its elevation will show its true length; but by making a projection of the planes of projection and the perpendicular on a plane at right angles to XY the perpendicular will then be projected on a plane parallel to it and will therefore have its true length shown in this new projection. In Fig. 360, X,Y, is at right angles to XY. a', is the new elevation of the point A and o'a', the point A from XY.

(1)

a+

FIG. 360.

is the true distance of

Having mastered the

159. Auxiliary Projections of Solids. rules for auxiliary projections of a point the student should have no difficulty in applying them to the drawing of auxiliary projections of solids. An example is illustrated by Fig. 361. The eleva tion (1) and plan (2) represent a right rectangular prism in a simple position. From the plan (2) an elevation (3) is projected on a ground line X1Y1. Considering the point R, the distance of r from X,Y, is equal to the distance

(3)

(2)

FIG. 361.

of from XY; also the straight line rr' is perpendicular to X,Y1. From the elevation (3) a plan (4) is projected on a ground line X,Y. Again considering the point R, the distance of r, from X,Y, is equal to the distance of r from X,Y,; also the straight line r'r, is perpendicular to X,Y2.

2

160. Projections of a Solid when a given Line in it is vertical. A plan and an elevation of the solid are first drawn when it is in a simple position such that the given line is parallel to the vertical plane of projection. The plan of the given line will be parallel to the ground line. A new ground line is then taken at right angles to the elevation of the given line and a new plan of the solid is projected from the elevation. This new plan will be a plan of the solid when the given line is vertical. Two examples are illustrated by Figs. 362 and 363.

In the first example (Fig. 362) the plan of a right hexagonal pyramid is required when an edge containing the vertex is vertical. The solid is placed with its base on the horizontal plane and with a sloping edge VA parallel to the vertical plane of projection; the plan va of

this sloping edge is parallel to XY. Completing the plan (1) the elevation (2) is projected from it. X,Y, is drawn at right angles to v'a' and from the elevation (2) the plan (3) is projected. Applying the rules for auxiliary projections of a point it is found that the new plans v1 and a, of the points V and A, coincide, which shows that the line VA is vertical when the plane of the plan (3) is considered to be a horizontal plane. An elevation (4) is also shown projected from the plan (3) on a ground line X,Y

In the second example (Fig. 363) it is required to draw the plan of a cube when a diagonal of the solid is vertical. The cube is first placed with one face on the horizontal plane. The plan of the cube in this position is a square which is drawn with one diagonal ac parallel to XY as shown at (1). The elevation (2) is projected from the plan (1). ac is the plan and a'c' is the elevation of a diagonal of the cube. XY, is drawn at right angles to a'c' and the plan (3) is projected from the elevation (2) as shown. The plan (3) is the one required. The outline of the plan (3) is a regular hexagon.

The plan of a solid may be drawn when a given line in it is inclined at any given angle to the horizontal plane by proceeding in the manner just explained, but making the new ground line inclined at the given angle to the elevation of the given line instead of perpendicular to it. But as the solid in this case can occupy any number of positions with reference to the horizontal plane and still fulfil the given condition, the problem is indefinite. In the case where the given line is vertical, however, the plan is always the same. These remarks will be understood if the solid be imagined to turn round the given line as an axis.

161. Projections of a Solid when a given Face is inclined at a given Angle, the Base of that Face being horizontal.The solid is first placed in a simple position and having the plan of the base of the given face at right angles to the ground line. The elevation of the given face will then be a straight line. A new ground line is then drawn, making with the elevation of the given face an angle

equal to the given inclination. A plan on this new ground line is the plan required.

Two examples are illustrated by Figs. 364 and 365. In the first example (Fig. 364) projections of a right hexagonal pyramid are determined when the face VAC is inclined at a given angle to the plane upon which the new plan is to be projected. The plan (1) and elevation (2) are drawn representing the pyramid with its base on the horizontal plane and with ac, the plan of the base of the face VAC, at right angles to XY. It will be seen that the elevation of the face VAC is the straight line 'a'. X,Y, is drawn at the given inclination O to v'a', and the plan (3) is projected as shown.

In the second example (Fig. 365) projections of a regular octahedron are determined when one face VAC is horizontal. The various projections are drawn in the order in which they are numbered. From the plan (3) an extra elevation has been projected on the ground line

XY. It is of interest to observe that the boundary line of the plan (3) is a regular hexagon, and this plan may evidently be drawn directly without the aid of any other projections.

Exercises XIV

1. Draw the elevations of the points given in Fig. 366 on a ground line inclined at 45° to XY.

2. Find the true distances of the points given in Fig. 366 from XY.

3. A triangle ABC is given in Fig 367. Draw an elevation of this triangle on a ground line parallel to ac. Also, project from the given elevation a new plan on a ground line parallel to a'c'.

4. A figure ABCDEF (not a plane figure) is given in Fig. 368. Draw an elevation of this figure on a ground line parallel to bc. Also, project from the given elevation a new plan on a ground line parallel to a'b'.

5. A plan and an elevation of a curved line are given in Fig. 369. Draw an elevation of this curve on a ground line inclined at 45° to XY.

6. Show the plans and elevations of the following points :-A, 1.5 inches in front of the V.P., above the H.P., and 2 inches from XY. B, 1.2 inches above