§ 60. General information on positional tasks

Tasks related to the solution of questions of the mutual arrangement of geometric shapes in a complex drawing are called positional.

Among the positional it is possible to distinguish two groups of tasks of the greatest practical interest. These include tasks for mutual belonging and tasks for mutual transfer. The tasks of the first group were repeatedly mentioned when studying chapters 7 and 8. This is explained by the fact that any line is a derivative of a point, and any surface is a derivative of a line. Specifically, the issues of belonging to a point of a straight line are considered in § 44, the belonging of a point and a straight plane in § 49, and the belonging of a point and a surface line in § 55.

The solution of positional problems of belonging involves working with lines of the surface graphically simple, for example, a straight line or a circle. This is necessary in order not to complicate the constructions in the complex drawing. To choose these lines correctly, you need to know which line families this or that surface carries.

Mutual intersection tasks are associated with the construction of points belonging simultaneously to two considered geometrical images, for example, a straight line and a plane, two planes, a plane and a surface, and two surfaces. Each of these points is built at the intersection of two auxiliary lines. These lines should be graphically simple and belong to one auxiliary plane or surface. The choice of auxiliary surfaces (intermediaries) that carry auxiliary lines depends on the shape of the intersecting surfaces. The combination of constructed common points allows you to build a line of intersection of geometric images.