# Intersection of curved surfaces, General information on the...

## General information on the intersection of curved surfaces

General Information. The shape of most of the most complex and responsible original parts of devices and machines is formed by the combination of various elementary bodies located in space so that their surfaces intersect each other. Therefore, an important stage in the construction of such details is the determination of the boundaries of elementary source surfaces, which are the lines of their mutual intersection.

We have already discussed the construction of lines of intersection of certain surfaces and bodies with each other: two planes (§ 4.2, 4.4), polyhedra (§6.6).

In this chapter, we consider the general method of constructing the intersection line of two curvilinear surfaces with each other, as well as some particular cases of intersection for different mutual disposition of surfaces and their position relative to the planes of projections.

The general way of constructing the line of intersection of two curved surfaces with each other. In the general case, the line of intersection of two curved surfaces is built with respect to the points that are found by means of auxiliary secant surfaces.

The two curvilinear surfaces P1 and P2 (Figure 10.1) are intersected by a third secant auxiliary surface p3. Find the intersection lines KL and MN of the auxiliary surface with each of the given ones. The intersection of the intersection lines constructed KL and MN belongs to the intersection line of the given surfaces.

Repeating such constructions many times, with the help of other auxiliary Fig. 10.1

The corresponding surfaces find the required number of common points of two surfaces for the line of their intersection.

We formulate the general rule for constructing the intersection line of surfaces:

- choose the kind of auxiliary surfaces;

- construct lines of intersection of auxiliary surfaces with given surfaces;

- find the intersection points of the constructed lines and connect them together.

As auxiliary surfaces, one chooses those whose intersection lines with given surfaces are projected onto a drawing into graphically simple lines - straight lines, circles. As auxiliary surfaces, for example, you can use planes or spheres. Let's consider their application.

Note that if one of the initial surfaces is ruled, then the problem of constructing the intersection line in this case can be reduced to the construction of the point of intersection of the straight line (the generator of the ruled surface) with the second given surface (see § 9.5).

When constructing, the methods of drawing conversion are applied, if this simplifies and refines the construction.

When constructing points of the intersection of surfaces, they first find those points that are called characteristic or reference points.

## Application of auxiliary secant planes

Consider the use of auxiliary secant planes by constructing the intersection line of a sphere with a cone of rotation (Figure 10.2).

For the construction of the intersection line of given surfaces, it is convenient to use as a secondary surface a series of horizontal planes perpendicular to the axis of the cone, which intersect the sphere and the cone along the circles. At the intersection of these circles find the points of the desired intersection line.

The construction begins usually with finding the projections of characteristic points. The projections 1 of the higher and 2 lower points are the points of intersection of the frontal projections of the sketches, since the center of the sphere and the axis of the cone lie in a plane parallel to the plane π2. Horizontal Г, 2 ' and profile 7 2 projections are in the projection connection. The projections 3, "3 ', 3 and 4 & quot ;, 4', 4 of the points lying on the equator of the sphere are found using the horizontal plane β (β ) passing through the center of the sphere O (O ). It crosses the sphere by eq Fig. 10.2

a rotor and a cone along a circle of radius r 2, in the intersection of the horizontal projections of which the horizontal projections 3 ', 4' of the points of the desired intersection line are found. The horizontal projections 3 ' and 4' of these points are the boundaries of visibility of sections of the intersection line on this projection. Intermediate point projections, for example 5, 5, 5, and 6, 6, 6 , are found using the auxiliary horizontal plane γ (y ). Their construction is clear from the drawing. Similarly, other points are constructed. The profile projections of points of the intersection line are plotted along their frontal and horizontal projections. Points with projections 7, V, 7 'and 8, 8, 8 are the boundaries of the visibility of sections of the profile projection of the intersection line. Below projections 7 and 8 the profile projection of the intersection line is visible. The exact construction of the projections of these points is considered in Fig. 10.5.

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