Projection 49

Since, in the revolution about the vertical axis X, each point has moved in a horizontal plane, it follows that the heights of points above the H plane have not been altered. Therefore, from the points of division on the ends of the cylinder in the plan, project for the elevation to the same level as the corresponding points in elevation A.

The construction is shown for the points 3 and 7 on one end, and for d and e on the other. This construction will give a series of points on each end of the cylinder. These points should be joined by the use of the irregular curve, pro ducing the two ellipses in the elevation.

For the last position, C, the cylinder must be revolved from the position of B without changing the angle of 45 degrees with V, until the final oblique position is reached. As the angle with V must not be changed, this second revolution must be made about an axis perpen dicular to V. The axis Y may be imagined to pass through any convenient point, as the ex treme left-hand point of Fig. B. In this revo lution (Article 64), the elevation does not change shape or size, but simply its position. Hence, in Fig. C, the elevation of B is drawn at the specified angle of 30 degrees with II, points 32, 72, and corresponding to the points 31, and of Fig. B.

The plan may now be obtained by project ing to meet T-square lines from the plan of B.

71. Auxiliary Planes of Projection. It has been shown in previous figures that often an end view must be made before constructing other views. Many times this end view may be made on H or V. Sometimes, however, it may be necessary or convenient to take an end view in a direction at right angles with neither V nor H.

72. Let it be required to draw the plan and elevation of a triangular prism as represented in Fig. 60.

Let the ends of the prism be equilateral triangles 5 inches on a side. The axis of the prism is to be 11 inches long, inclined at 30 de grees with V, backward and toward the right; the lowest edge of the prism is to be on H; and one face is to make 45 degrees with H. In this position, it is evident that a top view will not show the end at all, and a view taken in the usual direction for elevations will show the end, but not in its true size. If, however, a view be

taken in the direction of the arrow—that is, parallel to the direction of the axis—this view will be at right angles to the end of the prism, and the end will show in its real size and posi tion. To take a view in this direction is equivalent to projecting the prism on a new V plane parallel to the end of the prism, or—which is the same thing—perpendicular to the axis.

Hence, to make the construction, draw a line as in any convenient position perpendicu lar to the direction of the axis. This may be considered to represent the H plane seen edge wise when looking in the direction of the arrow. Then, with any point on H for the lowest corner, as 1', the end view of the prism is drawn showing the true size and slope, and one under face at 45 degrees with H. The width of the face 1-3-4-5 will appear in plan equal to the distance a-1'; and the width of face 1-2-6-5 equal to distance 1'-b, while the width of the upper face 2-3-4-6 will equal the length a-b.

With these widths and the given length of the prism, the plan might now be constructed without any further reference to the end view. It is, however, more convenient and more drafts man-like, to draw at any convenient distance from and parallel to it, a line 4-6 as one end; to project at right angles to a to 4, 5' to 5, and b to 6; then to draw from these points the edges 11 inches long and at right angles to the end 4-5-6, and finally draw the other end 3-1-2.

For the elevation, 1-5 is drawn on the H plane as the lowest edge, and directly above 1-5 in the plan. The height of 3-4 above H is seen in the end view as 3'-a, and the height of 2-6 as 2'-b; hence, in elevation, these edges will appear at these respective heights above the plane of 1-5. The distance 3-c is therefore made equal to 3'-a, and 2-d equal to 2'-b. In this elevation, the end 1-2-3, nearer the draftsman, is visible.

73. Summary of Important Principles. The important principles above brought out, may be summarized as follows: objects, or differences in height between different pa of the same object.