Seven planes whose horizontal traces are shown, and numbered from 1 to 7, will determine all the important points, and no other planes need be taken. Consider plane number 3. This plane cuts each cylinder in two straight lines whose plans are parallel to the plans of the axes of the respective cylinders, and whose horizontal traces are at the points where the horizontal trace of the plane cuts the horizontal traces of the cylinders. The elevations of these lines will be parallel to the elevations of the axes of the respective cylinders upon which they lie. These four lines intersect at four points in plan and in elevation and determine the four points on the intersection of the cylinders which have the number 3 attached to them. It will be observed that plane number 1 touches both cylinders. As already stated the horizontal traces of all the necessary cutting planes are shown, but the remaining construction lines are only shown for planes 1, 3, and 6, for the sake of making the figure clearer. The portion of the curve which is seen in plan, and which must therefore be put in as a full line, is found from the intersection of those lines which lie on the upper surface of one cylinder with those which lie on the upper surface of the other, that is, from the intersection of the lines whose horizontal traces lie on the part fgk of the ellipse with those whose horizontal traces lie on the part lmn of the circle. The remainder of the curve in plan will be hidden by one or other or both of the cylinders, and will therefore be put in as a dotted line. In like manner the part of the curve which is seen in the elevation is found from the intersection of the lines which lie on the front of one cylinder with those which lie on the front of the other, that is from the intersection of the lines whose horizontal traces lie on the halves of the horizontal traces of the cylinders which are furthest from XY. Instead of using the horizontal traces of the cylinders and the horizontal traces of the cutting planes, the vertical traces may be employed if the vertical traces of the cylinders come within a convenient distance on the paper. When one cylinder has a vertical trace but no horizontal trace within a convenient distance, and the other has a horizontal trace but no vertical trace within a convenient distance, it is necessary to use both the horizontal and vertical traces of the cutting planes. 322. Intersection of Cylinder and Cone. The general method of finding the intersection of a cylinder and cone is to cut the surfaces by planes parallel to the axis of the cylinder and passing through the vertex of the cone. These planes will cut the surface of the cylinder in straight lines parallel to its axis, and the surface of the cone in straight lines passing through its vertex. The intersection of the lines on the cylinder with the lines on the cone determine points on the intersection required. EXAMPLE 1 (Fig. 741). A right cone, base 2.9 inches diameter, axis 3.8 inches long, has its base horizontal. A cylinder 2 inches in diameter has its axis horizontal and 1.2 inches above the base, and 0.2 inch from the axis of the cone. It is required to draw the plan of the intersection of the surfaces and an elevation on a vertical plane inclined at 30° to the axis of the cylinder. The general method which has just been described can easily be applied to this example, but in this case the cutting planes may be horizontal, because horizontal sections of the cone are circles and horizontal sections of the cylinder are straight lines. In Fig. 741, horizontal cutting planes have been taken. Draw an elevation of the cylinder and cone on a plane perpendicular to the axis of the cylinder. In Fig. 741 this elevation is shown brought round into the plane of the other elevation. Consider the cutting plane of which L'M' is the vertical trace. This plane intersects the cone in a circle of which the diameter is equal to a 'b'. The plan of this circle is a circle concentric with the plan of the cone. This same cutting plane intersects the curved surface of the cylinder in two straight lines of which the points b,' and c,' are the end elevations. The plans of these lines are parallel to the plan of the axis of the cylinder, and at distances from it equal to half of b'c'. The points b and c where these lines on the plan meet the circle already mentioned, are the plans of points on the curve of intersection, and perpendiculars from b and c to XY to meet L'M' at b' and c' determine the elevations of these points. In like manner any number of points on the curve of intersection may be found. In Fig. 741 eight cutting planes are shown numbered from 1 to 8, and these should be sufficient. The most important points are on the planes 1, 3, 4, 5, 7, and 8. The position of the plane number 5 is obtained as follows. piq' is the vertical trace of a plane which touches the cylinder and passes through the vertex of the cone, p,' being the end elevation of the line of contact of this plane and the cylinder. op is of course perpendicular to Piqi. The vertical trace of plane number 5 passes through p. The lines in which the plane piq, cuts the cone are also shown and as these lines are tangents to the curve of intersection they assist in the correct drawing of the curve at important points. EXAMPLE 2 (Fig. 742). vv' is the vertex and va v'a' the axis of a cone whose vertical angle is 30°. be b'e' is the axis of a cylinder 2 inches in diameter. The dimensions which fix the positions of the axes of the cone and cylinder are given at the top right hand corner of Fig. 742. It is required to show, in plan and elevation, the intersection of the surfaces of the cone and cylinder. This example is worked by using cutting planes passing through the vertex of the cone and parallel to the axis of the cylinder which, as already stated, is the general method of finding the intersection of the surfaces of a cone and cylinder. The first step is to determine the horizontal traces of the cylinder and cone by Arts. 219 and 226 respectively. The construction lines for this are omitted in Fig. 742. All planes which are parallel to the axis of the cylinder and pass through the vertex of the cone will contain the line vt v't' drawn through parallel to be b'c'. Hence the horizontal traces of all the cutting planes will pass through t, the horizontal trace of vt v't'. There are eight planes required to determine all the important points but only three of these planes, numbered 1, 4, and 8, are shown in Fig. 742. Plane number 1 touches the cylinder and cuts the cone, and plane number 8 touches the cone and cuts the cylinder; each of these planes will therefore determine two points only on the intersection required. Plane number 4 cuts both surfaces and will determine four points on the intersection required. The omitted planes 2, 3, 5, 6, and 7 will each determine four points. 323. Intersection of Two Cones.-The general method in this 3 43 |