In the existing textbooks, deviations from the plan caused by central projection in aerial images have rather complicated expressions that discourage the reader. Limiting ourselves to the necessary minimum of such expressions, we have aimed to simplify them as much as possible by applying auxiliary quantities, usually hidden in the expressions, although giving a simple relationship to the quantities to be defined, similar to the way longitudinal parallax has greatly simplified the expression of heights. These auxiliary quantities we shall give in appropriate places.

In existing methods, the plane of the first image of the route is usually taken as the conditional horizontal plane. In this case, it is necessary to take into account the difference in flight altitude in the formulas expressing the distance. It seems simpler to us to take as a conditional horizontal plane the plane passing through the base and the line of intersection of the planes of a stereoscopic pair of adjacent planimetric aerial photographs of one route. We call this base plane the distance plane. The direction of the distances in each pair is slightly different from the vertical direction, since the difference in flight altitudes is small and the photographs are planimetric with a small angle of inclination. The lines of intersection of the planes of the images of a given pair with the distance plane will be parallel to each other and form straight lines of zero distortion of both images in relation to the distance plane.

Determining ordinates (deviations from target lines) according to existing instructions is quite difficult and not accurate enough, and therefore is not always used. It is advisable to offer something more rational. The position lines of angles close to 180° or 0° are almost parallel to the sides of these angles and their gradients are equal to the arithmetic sum or difference of the gradients of the sides of these angles. This makes it easy to use such position lines when obtaining the ordinates of the alignment of benchmarks set to determine surface displacement. These examples show the fruitfulness of the idea of using position lines. They are beginning to be introduced into practice in various areas, such as in matters of finding a lost center, staking out a project in the field, etc. It would be advisable to conduct experiments using the methods proposed here for determining displacements along alignments. It should be noted that these methods allow multiple measurements of angles and thereby increase their accuracy, whereas in other methods such an increase in accuracy is impossible without significantly complicating the measurement process.

The general theory of lines of position and gradients provides great clarity and simplicity in matters of graphical adjustment and determination of point position errors. In geodesy, when determining the position of unknown points on a plane, we directly measure horizontal angles and distances, which can be considered as functions of two variables (coordinates). To a given measured value of a function there corresponds a certain geometric locus of points on the plane—a certain line, which we shall call the line of position. Let us turn to the errors of point position. The position of the projection of a point onto the horizontal plane is determined by the intersection of two lines of position of two measured functions. Graphical adjustment using gradients is highly expedient in repeated trigonometric determinations of moving points, for example in landslide areas, in areas subject to displacement due to underground mining, and the like. Once constructed, the error figure with the calculated gradients will serve as a convenient means for further studies of the movement of the point being determined.