## Some special cases of intersection of surfaces

In some cases, the arrangement, shape or ratio of the dimensions of the curvilinear surfaces is such that no complex constructions are required to represent the line of their intersection. These include the intersections of cylinders with parallel generators, cones with a common vertex, co-axial surfaces of revolution, surfaces of revolution described around one sphere.

An image of the intersection of cylinders with parallel generators is shown in Fig. 10.10, on the left, cones with a common vertex - on the right.

** Co-axial surfaces of rotation. ** An image of the intersections of co-axially located different surfaces of rotation is shown in Fig. 10.11. The cone, intersecting with two cylinders of different diameters (a), is often used in the design as a transition from one diameter to another. The cone conjugated with a sphere, with the transition to cylinders (* b) *, is widely used as details of control mechanisms - handles.

A combination of three coaxial intersecting cones (c) is used in the design of parts called pins or rollers. The extreme conical surfaces, called chamfers, serve to strengthen the edge of the part and thereby prevent damage to the main working conical surface. The combination of intersecting two coaxial cones forms a center socket (d) for machining parts in the centers. The outer cone ρ1 is

Fig. 10.10

Fig. 10.11

To protect against damage to the working conic surface ρ2 when it comes into contact (impacts) with other parts.

** The intersection of surfaces described around one sphere ** (Figure 10.12). In this case, the lines of intersection of surfaces of the second order are two plane curves of the second order, represented on a plane parallel to the axes of the surfaces, in the form of a straight line -

Fig. 10.12

of the segments. Some examples of such intersections have already been given above (see Fig. 10.6, * in * and 10.7, * in. *

Other examples of the intersection of rotation surfaces described around one sphere are shown in Fig. 10.12. In the cases (* a *), (b) the surfaces of the two cylinders, the cone and cylinder intersect in two ellipses with the projections * 1 & 2; * and * 3 4 In the case (** in *) of the intersection of the cones with the projections ρ1 and ρ2 ", which have two parallel generators, the intersection lines are an ellipse with the projection * 1" 2 "and the parabola with the vertex at the point with the projection* * 3 *
The considered examples of the intersection of two surfaces of revolution described around one sphere are particular cases that follow from the Monge theorem: two second-order surfaces described near the third surface of the second order (or inscribed in it) intersect each other along two curves of the second order , the planes of which pass through a line connecting the points of intersection of the lines of tangency.

** Crossing surfaces when one of them is projecting ** (Figure 10.13). If one of the intersecting surfaces is projecting, then the problem of constructing the intersection line of two surfaces is simplified and reduces to constructing the missing projections of the curve of the line on one of the surfaces along one given projection of the line (see § 8.3). In Fig. 10.13. The horizontal projection of the line of intersection of a right circular cylinder and a sphere coincides with the horizontal projection of the cylinder. The front and profile projections of the line are constructed according to the sphere's affiliation by means of projections of auxiliary lines on the sphere. Note the characteristic (support) points of the intersection line, using the horizontal projection. Higher and

Fig. 10.13

the lowest points (their projections * 2, "2 ', 2' 'and* * 1,', 1, * ') lie in the plane of symmetry of the figure passing through center of a sphere with projections * O ", O '* and the axis of the cylinder with projections OiO1", * 0 [. * The horizontal projection of the plane of symmetry is a straight line passing through the projections O1HO11. At the intersection of this line with the projection of the cylinder, we mark the horizontal projections * 2 '** and* * 1' * of the highest and lowest points of the intersection line. Note that the point 2 is closest to the highest point of the sphere, and the point * 1 * is the most distant from it. The points * 3 * and * 4 - * are the left and right margins on the frontal and horizontal projections, their profile projections * 3, "4 '* - on the projections of generators , which coincide with the projection of the axis of the cylinder. Points 5 and * 6 * are on the main meridian of the sphere, their front projections * 5 * and * 6 - * on the frontal sketch of the sphere, profile * 5 * and * 6 * - on the profile projection of the vertical axis of the sphere. The points 7 and * 8 - * nearest to the plane π2 and farthest from it, their front projections 7 and * 8 * - on the projection of the axis of the cylinder, and profile * 7 * and * 8 ' * - on the extreme left and right projections of the generators. The points * 9 * and * 10 * have projections * 9 * and * 10 * on the front projection of the vertical axis of the sphere, the projection * 9 '* and * 10 - * on the profile projection of the sphere sketch.

The considered features of characteristic points make it easy to verify the correctness of constructing the intersection line of surfaces if it is constructed from arbitrarily chosen points. In this case, ten points are sufficient for smooth projections of the intersection line. If necessary, any number of intermediate points can be constructed.

The projection * 1 * of the lowest point is constructed using the projections of the parallel of the sphere. The highest point projection * 2 * is constructed using the projections of a circle of radius * O A * on the surface of a sphere whose plane is parallel to the plane π2. Similar constructions of the remaining projections of points of the intersection line are clear from the drawing.

The constructed points are connected by a smooth line, taking into account the peculiarities of their position and visibility.

## thematic pictures

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