Correlation

The task presented in this lesson is implemented in the environment of the Simplex software system.

Today we'll talk with you about this kind of projective transformation: about correlation.

Correlation is related (dual) to collineation, which we have already talked about, and it acts similarly, but it is set somewhat differently than the projective transformation we have already studied.

I hope you remember that collineation on a plane is determined by the correspondence of four pairs of points. The fact that collineation can be determined by the correspondence of four pairs of lines and even by a mixed correspondence of pairs, we have not yet said. We will correct this omission, but the most important thing to note now: these are pairs of homogeneous objects. In the three-dimensional case, collineation is specified by the correspondence of five pairs of points, in the four-dimensional six, etc.

Correlation will also be determined by the correspondence of pairs of geometric objects, but not homogeneous, but of the type of object - an object dual to it. Therefore, in the plane of its reference will be made up of pairs of the form point - line, in space - point - plane. We already observed a similar correspondence when we talked about polarization. Remember, in polarization relative to the conic, an arbitrary point on the plane uniquely passes into a straight line of this plane, and a straight line into a point. In three-dimensional space: a point in a plane and vice versa. This is the most important property of the polar transformation, and that is why they say that it is correlative.

Correlation will also transform a point into a straight line, and a straight line into a point. But she will convert a conic into a conic. I think that it will be interesting for all of you to see how the geometrical scheme of constructing a conic on five tangents in the blink of an eye turns into a scheme for constructing a conic on five points. Even more interesting (I think that no one has ever seen this scheme in view of the complexity of its manual implementation) is to see how the unknown to us scheme of constructing a quadric given by nine tangent planes looks in spatial correlation, translating this problem into a nine-point quadric construction scheme that we already know. This is already a very difficult task, which we still try to solve.

Define correlation in the plane. We draw its frame, which will consist of four points and four lines. Of course, each point must correspond to one of the lines. The correspondence is denoted by color (Fig. 1).

Fig. 1. Reference correlation conversion

As in the case of determining a collinear transformation, we construct two projective pencils of straight lines on the set of points. Let us choose, for example, the blue point as the first center of the bundle, and the green point as the center of the second bundle (Fig. 2 a, b).

Fig. 2 a. Plotting straight lines of the first bundle centered at a blue point

Fig. 2, b. Plotting the straight lines of the second bundle centered at the green point

On straight lines we will do something similar, but we will build not two pencils of lines, but two linear rows of points (these are dual objects with respect to pencils). I will write this construction in a little more detail.

We select the blue line (after all, it corresponds to the blue point) and find the points of its intersection with the red line (the intersection point is denoted with burgundy color), then with the purple line (the intersection point will be olive) and with the green line (the suppression point is blue-green) . It is very important to observe color matching (Fig. 3, a)!

Fig. 3 a. Determination of the first linear row of points corresponding to the first bundle

Now we perform similar actions with respect to the green line, since it corresponds to the green center of the second bundle. The blue-green dot we already have. At the intersection of the green line with the red, we get the orange point, and at the intersection with the lilac line, we get the blue point (Fig. 3, b).

Fig. 3, b. Determination of the second linear row of points corresponding to the second bundle

Now we have everything we need in order to set two projectivities between the corresponding bundles and rows.

The first projectivity will be defined by the carrier - the blue dot and three lines: burgundy, olive and blue-green, as well as the second carrier - the blue line and three points: burgundy, olive and blue-green.

The second projectivity will be defined by the carrier - the green dot and three lines: blue-green, orange and blue, as well as the second carrier - the green line and three points: blue-green, orange and blue.

We set some arbitrary point (black) in the drawing, draw two lines from it to the centers of the bundles of lines: light green and yellow (Fig. 4).

Fig. 4. The choice of an arbitrary point on the plane and the comparison of two direct bundles with it

We find the corresponding points in the projective correspondences for the light green and yellow lines - we get two points: light green on the blue line and yellow on the green line (Fig. 5).

Fig. 5. The result of the projective transformation of light green and yellow lines in the respective projectivities

It remains to draw a straight line over two points obtained and our transformation is ready (Fig. 6)!

Fig. 6. Correlative transformation of a point into a straight line

Suppose we now need to find the image of a straight line in the correlation. Draw such a line (black) (Fig. 7).

Fig. 7. The source line subjected to correlative transformation

We are now able to find correlative correspondence for points. We place two arbitrary points on a straight line and transform them into correlations using the same algorithm that we just examined.

We find the lines - the images of two selected points, intersect these lines and get the point - the image of the originally given line (Fig. 8).

Fig. 8. Building a point - an image of a straight line in a given correlative transformation

And of course, it's interesting to look at what will happen to the conic in the correlative transformation. The illustration demonstrates the transformation of a conic into a conic, and the tangent lines to the original conic were transformed into points through which the conic image passes.

Fig. 9. Correlative Conic Conversion

The following figure shows the result of a collinear transformation of the same tangents defining the original conic into tangents to the conic image. Collineation is defined using the diametric points of the curves. Note that the tangent-images went exactly through the points that we received from the original tangents in the correlative transformation.

Fig. 10. Illustration to explain the joint results of the correlative and collinear transformation of conics