Преобразование обратными радиусамиВ основе реципрочных проекций лежит реципрочное преобразова­ние, или так называемое преобразование обратными радиусами, которое вытекает из инволюции точек на прямой. В инволюции точек на пря­мой имеются две вещественные или две мнимые двойные точки, кото­рые делят гармонически любую пару со­ответственных точек. Средняя точка ме­жду двойными точками называется цен­тром инволюции и ей соответствует бес­конечно удаленная точка прямой. Произ­ведение расстояний двух соответствен­ных точек до центра инволюции есть величина постоянная.

Методы проективной геометрии дают возможность пользоваться, многочисленными геометрическими системами, что широко исполь­зовано в трудах Е. С. Федорова. В настоящее время даже в учебных руководствах по проективной геометрии не упоминается о существо­вании различных геометрических систем, которые, однако, открывают широкие возможности для решения ряда задач инженерной и научной практики.

In the author's paper “On One Regularity in the Location of Central Projections on the General Plane of a Picture", the proof of a theorem expressing the regularity of the relationship between the images of the elements of space obtained by projecting them from two different centers onto the general plane of the picture is given. The proved theorem is formulated as follows.

In the works of the author “Application of the method of rectangular coordinates to the solution of some problems of photogrammetry” and “About one regularity in the location of center projections on the common picture plane” the essential sides of the question of establishing regularities of connection between different in nature methods of projection adopted in descriptive geometry are stated.

In the course of the long historical development of descriptive geometry, as one of the sections of geometry, a number of questions have arisen that currently require scientific research. One of such questions is the question of establishing specific patterns of connection between different in nature methods of design. In graphically solving the various problems of engineering practice, it is necessary to strive for the simplest solution of each problem posed. In a number of cases, the simplest graphical solution of this or that problem can be achieved not only by skillful and reasonable use of one or another design method, but also by the ability to move from one design method to another by the simplest graphical constructions, i.e. to carry out transformation of projections. In the present work it is meant to establish a regularity of connection between the images of elements constructed by their central projection onto a common picture plane, and to show the possibility of using such a regularity in solving some problems of engineering practice.

In this paper, the solution to the problem under consideration is shown on the basis of the mutual connection between the methods of central and parallel projection. This mutual connection can be established by the method of rectangular coordinates in application to the named design methods. The method of rectangular coordinates quite definitely establishes a mutual connection between the different design methods - central and rectangular. The mutual connection allows, in matters of transforming central projections into rectangular ones, to use uniform simple relationships that take place when designing the main geometric image - a point. The use of these relationships is equally applicable to any case of the position of the plane of the picture relative to the coordinate system, which can be considered as a completely definite pattern. The transformation of central projections into rectangular ones can be carried out when solving photogrammetry problems related to determining the shape, size and position in space of objects from their photographs on a plane occupying any position relative to the surface of the earth.