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The radii wa, wß, wy..., drawn through these points of division, will represent the corresponding generatrixes oa, ob, 00... of the cone when developed.
The points of the conic section situated on these generatrixes will not change their respective distances from the vertex, and it is evident that these distances are given on the line o'a', between the point o' and the parallels to xy drawn through the projections é', f', g'; hence if these distances are set off from w to 4,0, X..., the curve e ox, representing the development of the elliptic section, can be drawn. The tangent is obtained as it was in the case of the cylinder, by observing that before and after the development, each element of the conic section always makes the same angle with the contiguous generatrix. Let the point v, for example, correspond to the point n, n' of the conic section. The tangent at this point is the hypotheneuse of a rightangled triangle, one side of which is mr, and the other, which has mn for its projection, is equal to uv; if, therefore, a perpendicular up be drawn to uv, and made equal to mr, and vp be joined, the line vp will be the tangent to the new curve egy....
When the cone is not a right one, the development cannot be made by means of a segment of a circle such as waa'. In this case, the simplest mode of proceeding is to divide the base abc... into very small parts, and to consider these as right lines, and the cone as a pyramid. The three sides of each triangular face of this pyramid is easily deduced, and they may be constructed successively, and the development of the cone thus obtained. It is equally easy to determine the distances from the vertex of the cone to the different points of the conic section, and consequently to trace the curve produced by the development of this section. The tangent is determined by constructing on y v an oblique
angled triangle, one side of which, up, is equal to mr, but the third side vp, must be deduced from its two projections nr and n"q.
Scholium. It is only for simplicity's sake, that, in the enunciation of this Problem, the cutting-plane is assumed, perpendicular to the vertical plane. But though the constructions requisite, when the plane is assumed in any other position, are more complicated, they are not more difficult.
PROBLEM 4. To determine the intersection of a cone with a plane when
the section has asymptotes. 50. The sectional plane pap' being again assumed perpendicular to the vertical plane, let a plane, o'm'm, be supposed parallel to it, passing through the vertex of the cone, and cutting this in the generatrixes om, o'm', and on, o'n'. The generatrixes indefinitely near these meet the plane pap', but at an indefinite distance; so that those which lie in the plane omm' must be considered as being met by pop'in points infinitely distant. Hence it appears that the curves of the section must have infinite branches. In other respects, all the constructions of the last problem must be employed in this case. In the figure the assumed cone is a right one, and the section is an hyperbola ; the true form and magnitude are determined by turning the sectional plane down on the vertical plane; the one conical surface is alone developed in the figure.
The principal object in this problem is to determine the asymptotes to the hyperbolas, or, in other terms, the tangents at points in the curves infinitely distant, which points may be considered as lying in the generatrixes parallel to the plane pap. Now, if we conceive a tangent to be drawn to
the curve anywhere, and the point of contact to be more and more distant, the limit to this tangent will be an asymptote, consequently these asymptotes may be determined on precisely the same principles as any other tangents.
The tangent at any point of the conic section is the intersection with the plane pap' of a plane tangential to the conical surface, touching it in the generatrix containing the point. Draw the tangents mr, ns to the base abc...; these will be the horizontal traces of the tangent planes, touching the cone in the generatrixes passing through the infinitely distant points of the curve; hence the points r and s, in which these traces meet pq, are common to the asymptotes. If, therefore, through r and s, parallels to the generatrixes om, on be drawn, they will be the horizontal projections of the asymptotes. These asymptotes to the curve, when drawn of its true magnitude and figure, as well as to the curve of the developed section, are easily determined from the foregoing by principles explained in the preceding problems.
§ 2. INTERSECTIONS OF CURVED SURFACES.
Intersection of two cylinders. 51. The auxiliary planes may be assumed parallel to the generatrixes of the two solids, in which case these can only be cut by the planes in lines coinciding with the generatrixes; hence the points of the intersection sought are determined by means of right lines only.
Through any point a,a', assumed at pleasure, draw two lines parallel to the generatrixes of the two given cylinders: determine the horizontal trace bc of the plane passing through these two lines. The traces of all the auxiliary planes will be parallel to bc (Prop. 17).
Let one of these traces cut the bases of the cylinders in d, e, f, and g, it is clear that the four generatrixes passing through these points will all lie in one plane; consequently the points in which the projections of these generatrixes cut each other, will be those of two points in the common intersections of the cylinders; h, i, k, l, will accordingly be points in the horizontal projection of this curve, and h'ik'l' their vertical projections.
If hh', ii', kk', Il be joined, these four lines will be perpendicular to the ground line.
By a repetition of these constructions as many points in the curve of intersection may be obtained as may be deemed necessary.
The tangent at any point of the curve of intersection
will be that of two planes, one tangential to each cylinder in the proposed point. Let the point n, n', for example, be
the intersection of two generatrixes, cutting the bases of the cylinders in m, m' and pop. Draw mq, pq tangents to these bases at m and p. These tangents will be the horizontal traces of the planes tangential to the cylinders in the two generatrixes through n; the line ng, therefore, drawn through n and q, the point common to these two traces, will be the horizontal projection of the intersection of the tangential planes, or will be that of the tangent sought; q' in