55. Point and line on surface

In general, the line may belong to the surface or not to belong. A line belongs to a surface if all its points belong to this surface (see Fig. 103, line l). An exception is the case when the line is represented by a straight line, and the surface is a plane. In this case, for a plane to belong to a plane, it is sufficient that at least two of its points belong to this surface (see § 49). The tasks of building lines belonging to a surface are part of the task of building lines of intersection of surfaces by a plane and the intersection of two surfaces, which are dealt with in §§ 63, 64.

If the line does not belong to the surface, then they intersect. The simplest case is the intersection with the surface of a straight line. The problem is solved by concluding the given line into a projecting plane and constructing the natural size of the section, from which it is easy to determine the point of entry and exit of the straight line. Problems of this type are considered in § 63.

Point may belong to the surface and not to belong. A point belongs to a surface if it lies on a line located on this surface. In fig. 104, to the point M belongs to a spherical surface, since it is located on the line of the circle / g 'lying on this surface. Points A and B also belong to the spherical surface, since they are located on the lines of the essay circles belonging to the spherical surface. Examples of the belonging of a point of a surface can be given in the case of a conical surface (point M in Fig. 104, a), a torus surface (point M in Fig. 105) and a surface of a more complex shape (point M in Fig. 103).

The problem of determining the belonging of a point to a surface is solved as follows. If projections of surface elements and points are given, it is necessary to draw a line belonging to the surface on one of the projection planes through a given point and construct a projection of this line on one plane of projections. If the second projection passes through the second projection of a point, the point belongs to the surface, if it does not pass, it does not belong.

This problem can be considered on the example of fig. 104, a. In the complex drawing, a conical surface is defined by outline lines. The point M of the horizontal and frontal projections is also set. Through the horizontal projection of the point we draw the horizontal projection h _{1 of the} circle belonging to the conical surface. Having constructed the frontal projection h _{2 of} this circle, we make sure that it has passed through the frontal projection of the point. This confirms that the point belongs to the conical surface.

This problem can be solved in another way. With the same source data, we make a projection of one of the generators through the front projection M of ₁ point. Having built the horizontal projection of the h generator, we see that it passes through the horizontal projection M _{1 of} point M, and this allows us to conclude that the point M belongs to the conical surface .

The principles of building points and lines on surfaces are the basis for building intersection lines, cuts, cuts, penetrations, etc., which determines the construction of complex geometric bodies, and as a result - parts, nodes, machines, buildings, structures.