§56. Overview of complex drawing conversion

In a complex drawing, geometric objects are projected in such a way that many of the elements that make them, for example, straight line segments, angles, and flat figures, are depicted with distortion. At the same time, when solving many problems, it is often necessary to transform a complex drawing so that the necessary element is located parallel or perpendicular to one of the projection planes. For example, it is necessary that the line segment, which is the edge of the polyhedron, or the polygon, which is the face of the polyhedron, should be located parallel to the plane of the projections. In this case, they are projected on this plane in full size.

The solution of such problems is greatly simplified if the elements of space that we are interested in occupy a particular position, that is, they are arranged parallel or perpendicular to the planes of the projections. The “degenerate” projections obtained in this case help to get an answer to the set task or to simplify the course of its solution. In order to achieve such an arrangement of geometric elements, the complex drawing is transformed or rearranged on the basis of specific conditions. Drawing transformation displays the change in the position of geometric images or planes of projections in space. The task of transforming a complex drawing can be solved by moving the projecting body in space to the desired position or changing in space the position of the planes of the projections relative to the geometric body. There are several methods for solving these problems. Methods of drawing transformation are mainly used: plane-parallel transfer, method of replacing projection planes (see § 36) and method of rotation.

Since there are two private positions for straight lines and two for plane, there are four initial problems for the transformation of a complex drawing:

• direct general position to make a straight level;
  
• make a level straight projecting;
  
• the plane of general position to make projecting;
  
• projection plane to make the plane level.