Descriptive geometry . ppes of the cone which arehere shown; but the upper nappes of the cones will intersectin the second branch of the same hyperbola. 195. Parallel Elements. If two elements, one in each sur-face, are parallel, the intersection of the surfaces may, thoughnot necessarily, have one or more infinite branches. XXV, § 195] CYLINDERS AND CONES 275 A. Two Cylinders. In the case of two cylinders, if thereis one pair of parallel elements, all the elements of both sur-faces are parallel, and the intersection can consist only ofstraight lines. B. A Cylinder and a Cone. In the case of a
Descriptive geometry . ppes of the cone which arehere shown; but the upper nappes of the cones will intersectin the second branch of the same hyperbola. 195. Parallel Elements. If two elements, one in each sur-face, are parallel, the intersection of the surfaces may, thoughnot necessarily, have one or more infinite branches. XXV, § 195] CYLINDERS AND CONES 275 A. Two Cylinders. In the case of two cylinders, if thereis one pair of parallel elements, all the elements of both sur-faces are parallel, and the intersection can consist only ofstraight lines. B. A Cylinder and a Cone. In the case of a cylinderand a cone, since no two elements of the cone are parallel,there can be but one element of the cone parallel to the ele-ments of the cylinder; but this element of the cone will thenbe parallel to all the elements of the cylinder. Hence, whenwe start to find the intersection by drawing a line through thevertex of the cone parallel to the elements of the cylinder,this line coincides with an element of the In Fig. 326, which shows an //-projection only, let A be thebase of the cone, and B the base of the cylinder, both bases inH. Let the //-trace of the line through the vertex of the coneparallel to the elements of the cylinder be at s, which is neces-sarily on the base of A, for reasons which have been the situation shown, since both limiting planes V and Zare tangent to the same base B, we might expect a two-curveintersection. One intersection is not affected b) the parallelelements, and is a closed curve, found by the elements 1 to .The second intersection does not exist, since the single elementthrough point .s- does not intersect the cylinder. Hence theintersection consists of a single closed curve. Now let the cone and cylinder !>e placed as in Fig. .>-7,which differs from Fig. 320 in that the plane M, tangent tothe cone along the parallel clement, lias now become ne ofthe auxiliary planes. The single intersection of the preceding 276 DESCRIPT
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