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One of the auxiliary planes touches the second cone along the line GW and cuts the first cone along the two lines OV, PV. In this case the four points are reduced to two, aa and BB', as may be seen from the figure. Again, there is a second tangent plane touching the second cone along DW and cutting the first cone, which also gives two points of the curves. These are called the limiting planes, for the curves of intersection must lie wholly between them.

To find the point situated on any given generator of either cone, take an auxiliary plane containing that line and find the point in the same way as before. For instance, to find the point on the generator WJ, that is the highest line of the cone, of which w'J is the vertical projection; TJ is the vertical trace of the plane containing that line, and determines the point X on the curve. It is important to find in this way the points on the contour or outline of the projections of the cone, for, as may be easily seen, these lines are tangents to the projections of the curve. Thus w'J is a tangent to the curve k'l'x' at the point x'. line Iw' touches the same curve at y'.

Similarly the

Remarks. Particular attention has been directed to the vertical projection of the curve in this problem, but the horizontal projection may be found in a similar way, by finding the points of intersection of the generators of the surfaces which are in the same plane, or when the projections of the generators intersect very obliquely, as the horizontal projections do in the figure, one projection of the curve may be found from the other by drawing the generators of one of the cones.

When the limiting planes both cut the same surface, as in this example, there are two curves of intersection. The smaller cone penetrates the larger and is completely enclosed by it, so that there is a curve of entrance and one of emergence. If both the limiting planes were not tangents to the same surface the two curves would merge into one another and form one curve.

As the tangent plane to the second cone along the line GW cuts the first cone along the generator PV, that genera

tor is the common section of the tangent planes to the two cones at the point of intersection of GW and PV, and consequently a tangent to the curve of intersection.

As the same may be said of the other points of the curve which are situated on the limiting planes, it may be stated generally, that the tangents to the curve of intersection at a point situated on a limiting plane is the generator in which that plane cuts the other cone.

PROBLEM II. Fig. 102.

To find the curve of intersection of a cone and a cylinder.

Let ABP be the horizontal trace of a cone, and V its vertex; CDO, the horizontal trace of a cylinder, and KL one of its generators; it is required to find the curve in which the two surfaces intersect.

The planes which cut the cylinder in straight lines, that is along generators, must be parallel to KL; while planes that cut the cone in straight lines pass through the vertex. Hence the auxiliary planes are to be taken passing through V and parallel to KL. In fact this problem may be considered as a particular case of Prob. I., the vertex of one of the cones being at an infinite distance.

Construction. Through the point V draw VT parallel to KL, and find its horizontal trace T. Now planes containing VT fulfil the required condition of cutting both surfaces in straight lines, so that any line through T, cutting the horizontal traces of the surfaces, may be taken as the horizontal trace of an auxiliary plane.

Draw any straight line AT cutting ABP in the points A, B, and CDO in the points C, D. The plane ATV cuts the cone along the two straight lines AV, BV, and the cylinder along the two straight lines CE, DF. AV meets CE in the point E, and DF in the point F; BV meets CE in the point G, and DF in the point H. Therefore E, F, G, H are four points in the curve required.

To find the points on the limiting planes, draw TK and TO tangents to CDO and cutting ABP at I, J, P, Q. These limiting planes touch the cylinder along the straight lines KL and OS, and cut the cone along the four lines IV, JV, PV and QV. The four limiting points of the curves are L, M, R and S. The lines IV, JV, PV and QV are tangents to the curves at the points L, M, R and S respectively, as has been shown in the preceding problem. To find the points of the curve which are on any given generator of either surface, draw the auxiliary plane containing that generator and proceed as before. For instance, to find the points on u'', draw the line TU cutting ABP at X and Y; the points of intersection of x'v' and y'v' with 'B' are the points where the curves meet u'ß.

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PROBLEM III. Fig. 103.

To find the curve of intersection of two cylinders.

Let ABO be the horizontal trace of a cylinder, and Pv, p'v' the projections of one of its generators; CDN the horizontal trace of a second cylinder, and Km, k'm' the projections of one of its generators; it is required to find the projections of the curve in which the two surfaces intersect.

As the section of a cylinder by a plane parallel to its generators is one or more straight lines, the auxiliary planes must be taken parallel to the generators of both the given surfaces.

Construction. From any point V of the generator PV draw VT parallel to KM, and find T its horizontal trace. Now the plane PVT is parallel to the generators of both cylinders (Theor. x. Chap. I.). Consequently the auxiliary planes are to have their horizontal traces parallel to PT.

Draw any straight line AD parallel to P cutting the horizontal trace of one cylinder in the points A, B, and the other in the points C, D. The points A, B, C and D are the horizontal traces of four generators, two on each surface, which lie in the same plane, so that their points of intersection E, F, G and H are four points of the required curve. In a similar way as many points as required may be found on the curve.

The limiting planes are JIL and NQS. As these planes are not both tangents to the same surface, neither cylinder is enclosed entirely within the other. In the figure one of the cylinders is supposed to be removed.

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