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.