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æy will be the vertical projection of q, and n'd' therefore that of the same tangent.
Of two cylinders which intersect, one may either entirely pass through the other, in which case there will be two curves of intersection, or a part only of the generatrixes of the one may meet the other cylindrical surface; when only one curve will be formed. The following construction will determine at once whether two given cylinders will intersect each other according to one or the other of these conditions,
Draw two tangents to the base of each cylinder parallel to bc. If both the tangents to either base cut the other, as ros, uv, the auxiliary planes, of which these lines are the traces, comprise between them the whole of the one solid, but only the portions of the second corresponding to the arcs egz, tfo, and it is obvious that the first cylinder meets these two portions of surface, and traverses the space between them ; in this case, therefore, there will be two curves of intersection.
But if each base be cut by one only of the tangents drawn to the other as is the case in the next fig. * ; then the auxiliary planes passing through the tangents rs and uz comprise the portions of surface of both cylinders corresponding to the arcs uro and szt, and it is evident that these portions mutually meet; while the remaining or the exterior portions have no common point; in this case there will therefore be only one curve of intersection.
The constructions above described for determining whether the cylinders meet in two, or only in one continuous curve determine the generatrixes of each solid, which serve as limits to the curve of intersection. Thus in the first fig., the
* Refer to the former fig. for the letters of reference not inserted in the second.
curve in which the one cylinder enters the other is bounded by the generatrixes which pass through the points t and v, and the other curve by which the cylinder passes out is bounded by the generatrixes which pass through s and z.
In the above fig. the curve of intersection has for its limits on one of the cylinders the generatrixes through u and y, and on the other those which pass through s and t.
The general construction for determining the tangent to these curves, also proves that these generatrixes are tangents to the curve of intersection; thus for example, let one of the planes, tangential to one of the two cylinders, pass through the point r and the other through t. The former coincides with the auxiliary plane, the trace of which is rt; it accordingly intersects the second in the generatrix which passes through t; this therefore is a tangent to the curve of intersection. The projections of this generatrix are consequently tangents to that of the curve.
Intersection of two Cones.
52. A series of planes passing through the two vertices of the cones will cut their surfaces on straight lines, the intersections of which will be points in the curve sought. The tangent at any point of the curve will be the intersection of the planes tangential to the two cones.
Let the given cones be those represented in the figure, let that with the least base be called, for distinction, the “ small,” and the other the “great cone.”
great cone." Let c be the horizontal trace of the line ab, a'b' joining the vertices. The horizontal traces of all the auxiliary planes will pass through this point c. Draw the tangent cd, ce to the base of the small cone, which will cut that of the larger in f, g, h, and i. All the auxiliary planes, the horizontal traces of which as cm are comprised between these tangents, cut both cones and will determine points common to their surfaces; but those planes, of which the traces are external to these two tangents, cannot cut both surfaces.
Hence the intersection of these cones is composed of two parts*, one situated on that portion of the larger cone corresponding to the arc frmg, and the other on that corresponding to the arc hi, but in the figure the constructions appertaining to the former are alone shown.
* The conical surface, it must be remembered, consists of two parts, one on each side of the vertex; only one of these is shown in the figure.
The points a, b are the horizontal projections of the vertexes, the lines ad and bf, ae and bg, are those of the generatrixes in which the cones are cut by the auxiliary planes which have cf, cg, for their horizontal traces, consequently the points t, u in which these respectively meet, will be the horizontal projections of two points in the curve of intersection. It must be observed that this curve being bounded on the larger cone by the generatrixes projected in bf, bg, will have these line for tangents, and consequently the projection of that curve will have bf, bg for tangents.
Let om be the trace of any auxiliary plane and let it cut the bases in k, l, and m; this plane will cut the conical surfaces in generatrixes having ak, al, bm for their projections. The intersections n, o of bm with ak, al will be points in the horizontal projection of the intersection of the surfaces, and by a repetition of these constructions as many points may be obtained as may be deemed necessary.
The other projection is obtained by constructing the vertical projections of the generatrixes determined by the auxiliary planes, and will be readily understood from the figure without further description.
Let it be required to determine the tangent to the curve of intersection at the point q, 1 that in which the generatrixes aq, br meet. Draw the tangents to the base ps, rs, which will obviously be the horizontal traces of planes tangential to the conical surfaces in the generatrixes in question ; draw qs through the point s in which the tangents meet; sq, s'll will be the horizontal and vertical projections of the tangent at the point qd.
If this construction were applied to the points t, ť', U, the lines bt, b't', bu, b'u' will be obtained, as, obviously, ought to be the case.
To determine the intersection of two surfaces of revolution,
the axes of which meet each other. 53. The construction would be simple enough in the case of two surfaces, the axes of which were parallel ; for in that