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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.

Fig. 229.

In Figure 228 the axis of piece B is not in the same plane as that of D, but to one side of it to the distance between the centre lines C, D, which is most clearly seen in the top view. In this case the process is the same except in the following points: In the side view the line w, corresponding to the line w in the end view, passes within the line x before the curve of intersection begins, and in transferring the lengths of the full lines b, c, d, e, f to the end view, and marking the arcs b', c', d', e', f', they are marked from the point w (the point where the centre line of B intersects the outline of A), instead of from the point x. In all other respects the construction is the same as that in Figure 227.

Fig. 230.

In these examples the axis of B stands at a right-angle to that of A. But in Figure 229 is shown the construction where the axis of B is not at a right-angle to A. In this case there is projected from B, in the side view, an end view of B as at B', and across this end at a right-angle to the centre line of B is marked a centre line C C of B', which is divided as before by lines d, e, f, g, h, their respective lengths being transferred from W as a centre, and marked by the arcs d', e', f', which are marked on a vertical line and carried by horizontal lines, to the arc of A as at i, j, k. From these points, i, j, k, the perpendicular lines l, m, n, o, are dropped, and where these lines meet lines p, q, r, s, t, are points in the curve of intersection of B with A. It will be observed that each of the lines m, n, o, serves for two of the points in the curve; thus, m meets q and s, while n meets p and t, and o meets the outline on each side of B, in the side view, and as i, j, k are obtained from d and e, the lines g and h might have been omitted, being inserted merely for the sake of illustration.

In Figure 230 is an example in which a cylinder intersects a cone, the axes being parallel. To obtain the curve of intersection in this case, the side view is divided by any convenient number of lines, as a, b, c, etc., drawn at a right-angle to its axis A A, and from one end of these lines are let fall the perpendiculars f, g, h, i, j; from the ends of these (where they meet the centre line of A in the top view), half-circles k, l, m, n, o, are drawn to meet the circle of B in the top view, and from their points of intersection with B, lines p, q, r, s, t, are drawn, and where these meet lines a, b, c, d and e, which is at u, v, w, x, y, are points in the curve.

Fig. 231.

Fig. 231 a.

It will be observed, on referring again to Figure 229, that the branch or cylinder B appears to be of elliptical section on its end face, which occurs because it is seen at an angle to its end surface; now the method of finding the ellipse for any given degree of angle is as in Figure 231, in which B represents a cylindrical body whose top face would, if viewed from point I, appear as a straight line, while if viewed from point J it would appear in outline a circle. Now if viewed from point E its apparent dimension in one direction will obviously be defined by the lines S, Z. So that if on a line G G at a right angle to the line of vision E, we mark points touching lines S, Z, we get points 1 and 2, representing the apparent dimension in that direction which is the width of the ellipse. The length of the ellipse will obviously be the full diameter of the cylinder B; hence from E as a centre we mark points 3 and 4, and of the remaining points we will speak presently. Suppose now the angle the top face of B is viewed from is denoted by the line L, and lines S', Z, parallel to L, will be the width for the ellipse whose length is marked by dots, equidistant on each side of centre line G' G', which equal in their widths one from the other the full diameter of B. In this construction the ellipse will be drawn away from the cylinder B, and the ellipse, after being found, would have to be transferred to the end of B. But since centre line G G is obviously at the same angle to A A that A A is to G G, we may start from the centre line of the body whose elliptical appearance is to be drawn, and draw a centre line A A at the same angle to G G as the end of B is supposed to be viewed from. This is done in Figure 231 a, in which the end face of B is to be drawn viewed from a point on the line G G, but at an angle of 45 degrees; hence line A A is drawn at an angle of 45 degrees to centre line G G, and centre line E is drawn from the centre of the end of B at a right angle to G G, and from where it cuts A A, as at F, a side view of B is drawn, or a single line of a length equal to the diameter of B may be drawn at a right angle to A A and equidistant on each side of F. A line, D D, at a right angle to A A, and at any convenient distance above F, is then drawn, and from its intersection with A A as a centre, a circle C equal to the diameter of B is drawn; one-half of the circumference of C is divided off into any number of equal divisions as by arcs a, b, c, d, e, f. From these points of division, lines g, h, i, j, k, l are drawn, and also lines m, n, o, p, q, r. From the intersection of these last lines with the face in the side view, lines s, t, u, t, w, x, y, z are drawn, and from point F line E is drawn. Now it is clear that the width of the end face of the cylinder will appear the same from any point of view it may be looked at, hence the sides H H are made to equal the diameter of the cylinder B and marked up to centre line E.

Fig. 232.

Fig. 233.

It is obvious also that the lines s, z, drawn from the extremes of the face to be projected will define the width of the ellipse, hence we have four of the points (marked respectively 1, 2, 3, 4) in the ellipse. To obtain the remaining points, lines t, u, v, w, x, y (which start from the point on the face F where the lines m, n, o, p, q, r, respectively meet it) are drawn across the face of B as shown. The compasses are then set to the radius g; that is, from centre line D to division a on the circle, and this radius is transferred to the face to be projected the compass-point being rested at the intersection of centre line G and line t, and two arcs as 5 and 6 drawn, giving two more points in the curve of the ellipse. The compasses are then set to the length of line h (that is, from centre line D to point of division b), and this distance is transferred, setting the compasses on centre line G where it is intersected by line u, and arcs 7, 8 are marked, giving two more points in the ellipse. In like manner points 9 and 10 are obtained from the length of line i, 11 and 12 from that of j; points 13 and 14 from the length of k, and 15 and 16 from l, and the ellipse may be drawn in from these points.

It may be pointed out, however, that since points 5 and 6 are the same distance from G that points 15 and 16 are, and since points 7 and 8 are the same distance from G that points 13 and 14 are, while points 9 and 10 are the same distance from G that 11 and 12 are, the lines, j, k, l are unnecessary, since l and g are of equal length, as are also h and k and i and j. In Figure 232 the cylinders are line shaded to make them show plainer to the eye, and but three lines (a, b, c) are used to get the radius wherefrom to mark the arcs where the points in the ellipse shall fall; thus, radius a gives points 1, 2, 3 and 4; radius b gives points 5, 6, 7 and 8, and radius c gives 9, 10, 11 and 12, the extreme diameter being obtained from lines S, Z, and H, H.

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