In projecting, the lines in one view are used to mark those in other views, and to find their shapes or curvature as they will appear in other views. Thus, in Figure 225a we have a spiral, wound around a cylinder whose end is cut off at an angle. The pitch of the spiral is the distance A B, and we may delineate the curve of the spiral looking at the cylinder from two positions (one at a right-angle to the other, as is shown in the figure), by means of a circle having a circumference equal to that of the cylinder.

The circumference of this circle we divide into any number of equidistant divisions, as from 1 to 24. The pitch A B of the spiral or thread is then divided off also into 24 equidistant divisions, as marked on the left hand of the figure; vertical lines are then drawn from the points of division on the circle to the points correspondingly numbered on the lines dividing the pitch; and where line 1 on the circle intersects line 1 on the pitch is one point in the curve. Similarly, where point 2 on the circle intersects line 2 on the pitch is another point in the curve, and so on for the whole 24 divisions on the circle and on the pitch. In this view, however, the path of the spiral from line 7 to line 19 lies on the other side of the cylinder, and is marked in dotted lines, because it is hidden by the cylinder. In the right-hand view, however, a different portion of the spiral or thread is hidden, namely from lines 1 to 13 inclusive, being an equal proportion to that hidden in the left-hand view.

Fig. 226.

The top of the cylinder is shown in the left-hand view to be cut off at an angle to the axis, and will therefore appear elliptical; in the right-hand view, to delineate this oval, the same vertical lines from the circle may be carried up as shown on the right hand, and horizontal lines may be drawn from the inclined face in one view across the end of the other view, as at P; the divisions on the circle may be carried up on the right-hand view by means of straight lines, as Q, and arcs of circle, as at R, and vertical lines drawn from these arcs, as line S, and where these vertical lines S intersect the horizontal lines as P, are points in the ellipse.

Let it be required to draw a cylindrical body joining another at a right-angle; as for example, a Tee, such as in Figure 226, and the outline can all be shown in one view, but it is required to find the line of junction of one piece, A, with the other, B; that is, find or mark the lines of junction C. Now when the diameters of A and B are equal, the line of junction C is a straight line, but it becomes a curved one when the diameter of A is less than that of B, or vice versa; hence it may be as well to project it in both cases. For this purpose the three views are necessary. One-quarter of the circle of B, in the end view, is divided off into any number of equal divisions; thus we have chosen the divisions marked a, b, c, d, e, etc.; a quarter of the top view is similarly divided off, as at f, g, h, i, j; from these points of division lines are projected on to the side view, as shown by the dotted lines k, l, m, n, o, p, etc., and where these lines meet, as denoted by the dots, is in each case a point in the line of junction of the two cylinders A, B.

Fig. 227.

Fig. 228.

Figure 227 represents a Tee, in which B is less in diameter than A; hence the two join in a curve, which is found in a similar manner, as is shown in Figure 227. Suppose that the end and top views are drawn, and that the side view is drawn in outline, but that the curve of junction or intersection is to be found. Now it is evident that since the centre line 1 passes through the side and end views, that the face a, in the end view, will be even with the face a' in the side view, both being the same face, and as the full length of the side of B in the end view is marked by line b, therefore line c projected down from b will at its junction with line b', which corresponds to line b, give the extreme depth to which b' extends into the body A, and therefore, the apex of the curve of intersection of B with A. To obtain other points, we divide one-quarter of the circumference of the circle B in the top view into four equal divisions, as by lines d, e, f, and from the points of division we draw lines j, i, g, to the centre line marked 2, these lines being thickened in the cut for clearness of illustration. The compasses are then set to the length of thickened line g, and from point h, in the end view, as a centre, the arc g' is marked. With the compasses set to the length of thickened line i, and from h as a centre, arc i' is marked, and with the length of thickened line j as a radius and from h as a centre arc j' is marked; from these arcs lines k, l, m are drawn, and from the intersection of k, l, m, with the circle of A, lines n, o, p are let fall. From the lines of division, d, e, f, the lines q, r, s are drawn, and where lines n, o, p join lines q, r, s, are points in the curve, as shown by the dots, and by drawing a line to intersect these dots the curve is obtained on one-half of B. Since the axis of B is in the same plane as that of A, the lower half of the curve is of the same curvature as the upper, as is shown by the dotted curve.