252 FOCI AND DIRECTRIX. The point n' travels in elevation over the arc n'N', and the plan of N' is simultaneously on ab and on N'N perpendicular to xy; and the point n", the plan of which is b, similarly reaches the horizontal plane at N1. The required curve of intersection is an ellipse having NN1 for major axis, and for minor axis a length equal to the diameter of the cylinder. The minor axes of all ellipses which can be cut from the same cylinder are consequently of equal length, but the length of the major axis depends jointly on the diameter of the cylinder and the inclination of the cutting plane to its axis, since n'n" = a'b' cosec 0. Just as in the case of the cone, if spheres be inscribed in the cylinder touching the plane of section they will do so in the foci of the curve of intersection. The elevations of these spheres are the circles shewn in the figure touching the line lď in the points f' and f', and also touching a'a", b'b". f travels over the circular are f'F", and FF perpendicular to ay, meeting ab in F, determines F, one of the foci. The horizontal planes through the circles of contact of the spheres and cylinder intersect the plane of section in the directrices of the curve. d' is therefore the elevation of one of them, which after rotation of the section plane round Im comes into the position DX. Proof. The line whose elevation is f'r' is a tangent to the inscribed sphere, since it lies in a tangent plane to that sphere (the given section plane) and passes through the point f in which the sphere touches the plane. It is therefore equal in length to any other tangent to the sphere drawn from the point whose elevation is r', and since r' is really on the surface of the cylinder, the length of the tangents drawn from it to the sphere must be r'k', where r'k' is parallel to the axis of the cylinder, and k' is on the circle of contact of sphere and cylinder. But r'k' : r'd′ in a constant ratio = cos 0, and r'k' = F1R; r'd' = RM, where RM is perpendicular to DX meeting it in M, therefore FR: RM in a constant ratio, or the locus of R is an ellipse. THE OBLIQUE CYLINDER. DEF. If a straight line, which is not perpendicular to the plane of a given circle, move parallel to itself, and always pass through the circumference of the circle, the surface generated is called an oblique cylinder. The line through the centre of the circular base parallel to the generating lines is the axis of the cylinder. The section of the cylinder made by a plane containing the axis and perpendicular to the base is called the principal section. The section of the cylinder by a plane perpendicular to the principal section, and inclined to the axis at the same angle as the base, is called a sub-contrary section. It is evident that any section by a plane parallel to the axis consists of two parallel lines, and that any section by a plane parallel to the base is a circle. PROBLEM 134. To determine the sub-contrary section of an oblique cylinder. Let o (fig. 137) be the centre of the circular base, and the circle on ab as diameter the base of the cylinder; let ob be the plan of the axis. Draw xy parallel to ab, so that the elevation on xy as ground line will be parallel to the principal section of the cylinder; draw aa', oo', bb' perpendicular to xy meeting it in a', o', b', which will be the elevations of the corresponding points of the base. Since the elevation is parallel to the principal section, the angle which the elevation of the axis (i.e. the line o'c') makes with the ground line will be the real angle which the axis itself makes with the horizontal plane. Draw a'a,, b'b,' parallel to o'c', these lines are the elevations of the bounding lines of the solid projected on the vertical plane standing on xy. Draw any line ab'l making the same angle with o'c' as o'c' makes with xy, meeting xy in l, and draw Im perpendicular to xy. Im will be the horizontal trace and la,' the vertical trace of a plane of sub-contrary section; and if this plane be rotated round Im till it coincides with the horizontal plane, every point on the 254 SUB-CONTRARY SECTION IS A CIRCLE. surface of the cylinder between a and b,' will evidently reach a point on the circle on ab as diameter, i. e. the true form of the sub-contrary section is a circle. 1 The horizontal projection of the sub-contrary section is the ellipse having cc, projected from c', the point in which a,'b,' intersects the axis of the cylinder as major axis, and a,b, the projection of a,b,' as minor. 1 1 PROBLEM 135. To determine the section of an oblique cylinder by a plane not parallel to the axis, to the base, or to a sub-contrary section. Case I. Let the plane of section be perpendicular to the principal section (fig. 137). The horizontal trace (de) of the plane of section must be drawn perpendicular to ab, the plan of the axis. If the plane of section makes an angle (p) with the horizontal plane, the vertical trace must be drawn through d making this angle with xy. Let it meet a'a, in h' and b'b' in k'. Draw any circular section, as p'q', between h' and ' meeting dk' in r'; the plan is of course the circle on pq as diameter pro 1 jected from p' and q' on ab, and if the projection of the point r' cuts this circle in r and r, these will be the plans of two points on the required curve. If rr, meet pq in n we have rn2 = pn . nq. If now the plane of section be rotated round de till it coincides with the horizontal plane, h' travels in elevation to H' and in plan to H, k' travels in elevation to K' and in plan to K, and r and r, reach R and R, respectively. Therefore if RR, meet ab RN2 = pn. nq = p'r' . r'q' ; in N, 1 1 but p'r' : h'r' in a constant ratio, and r'q' : r'k' in the same ratio, p'r'. r'q' : h'r'. r'k' in a constant ratio; .. RN2 : HN. NK in a constant ratio, or the locus of R is an ellipse (Prop. 4, p. 108). Case II. Let the plane cut the cylinder in any manner (fig. 138). Let ab be the diameter of the base perpendicular to the horizontal trace of proposed section plane. The circle on ab is the plan of the base of the cylinder, ov the plan, and o'v' the elevation of its axis, the elevation being projected on a plane perpendicular to the proposed section plane. Lines through a and b parallel to ov are of course the plans of the generators through a and b, and if a and b are projected on to the ground line at a' and b' lines through these points parallel to o'v' will be the elevations of these generators and will be the bounding lines of the solid as seen in the proposed elevation. Im is the horizontal trace, and In,' the vertical trace of the section plane; let a'n' parallel to o'v' meet In,' in n', and b'n,' meet it in n'; n' and ' are evidently points on the required curve of intersection, and their plans n and n, are found by projecting n' and n,' on to the plans of the generators through a and b. Take any horizontal section of the cylinder between n' and n‚', as p'q'; the plan is of course a circle of diameter p'q', and its position can be determined by projecting p' and q' on to the plans of the generators through a and b, as at p, q. This horizontal section and the proposed section plane intersect in a line the elevation of which is r', the point in which p'q' cuts ln'', and the plan of this line cuts the circle pq in points r and r, projected from r', which are plans of points on the required curve of intersection. Now rotate the section plane round lm, its horizontal trace, till it coincides with the horizontal plane: in elevation the points n', r', n' travel over circular arcs to N', R', N'; in plan n, r, r1, n1 travel over lines perpendicular to lm to N, R, R,, N1, obtained by projecting N', R' and N'. 1 These are points situated on the true outline of the curve of intersection, and any additional number of points can be obtained in precisely the same manner. The curve is an ellipse having NN1 as a diameter and RR, as a corresponding double ordinate, so that DD1, the diameter conjugate to NN,, can at once be drawn 1 |