Descriptive geometry . und 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 DESCRIPTIVE GEOMETRY [XXV, § 195 case now contains two infinite points, numbers 3 and 7. Theresult is that the curve of intersection
Descriptive geometry . und 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 DESCRIPTIVE GEOMETRY [XXV, § 195 case now contains two infinite points, numbers 3 and 7. Theresult is that the curve of intersection, as traced on the sur-face of the cylinder, consists of two infinite branches, to whichthe elements 3 and 7 are asymptotic. These two branches ofthe intersection lie on opposite nappes of the cone. The preceding two examples do not exhaust all the possi-bilities of this case. They are, however, the most generalones, and, taken together, give a typical illustration of theformation of infinite C. Two Cones. In the case of two cones, to discover thenumber of parallel elements, draw a third cone, whose vertexcoincides with the vertex of one of the given cones, and whoseelements are parallel respectively to the elements of the , corresponding to the number of possible points whichtwo ellipses may have in common, we may find one, two,three, or four pairs of parallel elements, or every element ofone cone may be parallel to an element of the other latter case is that already illustrated in Fig. 325, theintersection being a hyperbola. The possibilities of the othercases are quite numerous. As an extreme result, each curveof a normal two-curve intersection may be broken into twoinfinite parts, so that the complete intersection consists of fourinfinite branches, two branches on each nappe of each the method of dealing with infinite points has alreadybeen shown for the case of the cylinder and cone, a furtherdiscussion will not be made. XXV, § 19
Size: 1733px × 1442px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometrydescriptive
