In the previous lesson, we examined the correlation transformation on the plane, which is defined by four point-to-line pairs. This transformation allows you to establish a one-to-one correspondence between points and straight lines of the plane.

A similar transformation naturally exists in spaces of higher dimensions. The set of linear images that participate in this transformation becomes richer. And today we will consider how the correlative transformation in three-dimensional space is defined and implemented using a plane-projection model - Monge diagram.

Obviously, in three-dimensional space, in the correlation, the point will correspond to the plane, and the plane to the point. A straight line located in three-dimensional space must unambiguously correspond to another straight line. Here the image transforms into a homogeneous image, like a collinear correspondence.

On the diagram we set five (red, blue, green, lilac and orange) points and five planes (red, blue, green, lilac and orange) (Fig. 1). Points and planes of the same color correspond to each other and determine the correlative transformation of three-dimensional space into itself.

Fig. 1. The benchmark of the correlative transformation in three-dimensional space

Draw a straight line through the red and blue points, and also determine a straight line from the intersection of the red and blue planes (Fig. 2).

Fig. 2. The choice of the center of the beam of planes in the first field and the corresponding center of a number of points in the second field of the correlation transformation

In the first field of the correlative transformation, we build three planes, connecting the blue line with the green, lilac and orange points (Fig. 3). We get the green, lilac and orange plane.

Fig. 3. The definition of the green, lilac and orange planes

Determine the intersection points of the obtained planes with a light green straight line obtained from the intersection of the planes of the second field (Fig. 4). We get the green, lilac and orange dots on a light green line.

Fig. 4. Obtaining the intersection points of the planes of the first field with a light green straight line

Now we determine the intersection points of the corresponding planes of the second correlation field with the same light green straight line (Fig. 5). We get green, lilac and orange dots on a light green straight line, marked with square markers.

Fig. 5. Obtaining the intersection points of the planes of the second field with a light green straight line

Knowing three pairs of points on one straight line, we can determine the projectivity of a point series. What we will do (Fig. 6).

Fig. 6. Assignment of projective correspondence on a straight line located in three-dimensional space

Now, as the center of the beam of planes in the first field, we take a straight line (blue) drawn through the red and green points. In the second field we determine a straight line (light green) of the intersection of the red and green planes (Fig. 7).

Fig. 7. The construction of the center of the beam of planes and the carrier of a point series for the implementation of the second projectivity

The obtained points determine the second projectivity on the second line located in three-dimensional space (Fig. 8).

Fig. 8. Setting the projective correspondence on the second straight line located in three-dimensional space

We perform a similar procedure with the third set of planes (Fig. 9).

Fig. 9. Preparation of data for determining the determinant of the third projectivity

Using the same scheme, we construct points and obtain projectivity (Fig. 10).

Fig. 10. Obtaining a reference point of projectivism on the third line

Let now we are given a point X, which we need to translate into a plane in a given correlation (Fig. 11).

Fig. 11. Defining an arbitrary point X

Connect the point X with the blue line - the center of the first beam of planes and determine the intersection point of the resulting plane with the first carrier of the linear point series (Fig. 12).

Fig. 12. Determining the intersection point of the first plane defined by the point X and the carrier line of the first bundle of planes

We will perform the same construction by connecting the point X with the direct-cent of the second beam of planes to obtain the desired point on the second carrier of the linear series of points (Fig. 13).

Fig. 13. Determination the intersection point of the second plane defined by the point X and the carrier line of the second pencil of planes

Determination of the third point is similar to the construction of the first two ones (Fig. 13).

Fig. 13. Three pairs of points obtained from the intersection of the planes induced by the point X

So, we have three points that need to be transferred to the second field of correlation in the respective projectivities. The result of the transformation is three points that define the plane. This plane is the image of the point X in the correlation transformation specified by the original frame. The problem is solved (Fig. 14).

Fig. 14. Plane - the image of point X in a given three-dimensional correlation