If you look through other similar books or "walk" through mathematical sites that talk about second-order curves, then you will see the same thing: a hyperbola has a hyperbole conjugated to it. The question is, if something like this does not exist in the ellipse, because it does not have an imaginary diameter: both diameters are real. At the same time, we know that from a projective point of view, hyperbole and ellipse do not differ. This means that if a hyperbola has a conjugate curve - a hyperbola, then the ellipse must also have something conjugate. And the imaginary diameter, apparently, should be the reason. Let's try to figure out where there is a discrepancy.

Perhaps I already wrote earlier that the concept of "imaginary diameter" of a hyperbola is a wrong concept. Now I can't find in the records where I talked about this, therefore I will state my position again, even if you know it.

The feeling that a hyperbola has an imaginary diameter stems from the analytical form of writing the hyperbole equation in comparison with an ellipse. These equations are based on affine representations of second-order curves in which there is no concept of infinity. It turns out that a hyperbola is an open curve with two separate branches, and an ellipse is a closed curve with one branch.

Projective geometry claims (and rightly so) that a hyperbola is a closed curve with one branch, which, unlike an ellipse, intersects an infinitely distant line at two improper points. Moreover, any non-degenerate curve of the second order — a conic — can be collinearly completely transformed into another non-degenerate curve by a collinear transformation. It is only necessary to find such a transformation. What we will do now. And then we'll see how, for example, the diameters and tangents of various affine types of curves will behave under such transformations.

New Project

We define two ellipses on the plane.

Fig. 5. The result of the transformation of a point series, with support on the first conic to a point series on the second conic - a hyperbola

From the result it should be concluded that the second conic - the hyperbola - has a legitimate second main diameter and this diameter is an infinitely distant straight line. This diameter is not imaginary, but real!

It follows from this that the definition of the center of a second-order curve, which supposedly is the image of an infinitely distant straight line in polarization, is incorrect! The center of the second-order curve lies at the intersection of its main diameters, so the center of the hyperbola is not at the point "lying" in the middle between the "hyperbola segments", where they approach each other at the minimum distance, but at infinity in the direction of the first main diameter of the curve!

For the same reason, the concept of the diameter of a second-order curve in general is incorrectly interpreted. All diameters must go through the center of the curve, and if the center of the hyperbola is at infinity, then all diameters must go through this infinitely distant point, that is, they are all parallel to the first main diameter!

The foregoing can be easily illustrated.

Fig. 6. On the construction of diameters of a hyperbola

Fig. 7. Conversion of the conic asymptotes into lines tangent to an ellipse at the intersection points of one of the diameters with the curve

It can be seen from the drawing that the asymptotic lines were transformed into straight lines tangent to the ellipse at the points of intersection of the same diameter of the ellipse with itself.

But then we still have two unused points from the second diameter. Having drawn two tangents to the ellipse through these points, we transform them into a direct collineation.

Fig. 8. Collinear transformation of blue tangents drawn to the left curve into tangents obtained from the right curve

The result is not long in coming. We see that the lines turned into two tangents to the hyperbole at the points of intersection of the first diameter with the curve itself.

Everything is strict, harmonious and logical!

And now we can talk about tangent hyperbolas.

To begin with, we construct an arbitrary hyperbola and find it conjugate as described in the reference book. We set the hyperbola.

Fig. 9. The initial hyperbola in the problem of constructing the conjugate hyperbola

Fig. 10. The construction of the "main" tangents to the hyperbole.

Fig. 11. Construction of four intersection points of the main tangent conics

Fig. 12. Determination of the elements of the frame of the conjugate hyperbola

Fig. 13. Blue hyperbole - the desired conjugate hyperbole

Fig. 14. Constructing a conjugate hyperbola to an ellipse

Based on the results obtained, it should be concluded that the definition given in the reference manual with respect to hyperbola does not have generality and, moreover, is not based on those geometric images that are the root cause of this geometric phenomenon.

It would seem that we have already received a new result. But still, we will not stop. Perform another collinear transformation, but this time translate the red ellipse into a new red circle. We will transform in the new collineation the second blue hyperbola.

Fig. 15. The result is the transformation of conics into a circle and the form of a conjugating hyperbola for a circle

Fig. 16. A hyperbola conjugate to a circle

Fig. 17. The second conjugate conic to the circle

Fig. 18. Construction of the second conjugate conics

The algorithm presented in our conversation convinces us that any conic has two, and not one, main conjugate conics.

New Projects

Having the opportunity to build conjugate conics, we begin the study of their properties as a single geometric design. Imagine a family of three conjugate conics in Fig. 19.

Fig. 19. Initial data for explaining the geometric properties of conjugate conics

Property 1

The polarity induced by any of the conics of the resulting construction translates both conics conjugate to it into themselves.

Fig. 20. The result of the conversion of the red conic in polarization relative to the green conic

Property 2

Four real foci of two hyperbolas included in the conjugation complex are on the same circle.

Fig. 21. Illustration to the explanation of properties 2, 3 and 4

Property 3

The circle passing through the real foci intersect one hyperbola at the points through which the directrixes of the second hyperbola pass (see Fig. 21).

Property 4

Directrixes of adjacent hyperbolas intersect at points lying on an ellipse conjugate to these hyperbolas.

Property 5

The circle drawn through the foci of the ellipse and one of the foci from the hyperbola that does not lie with the original two on the same line intersects the ellipse at the points obtained from the intersection of the directrix corresponding to this focus with the ellipse, and the center of this circle lies on the directrix. This statement is true for both real and imaginary foci. Such a circle will be called focal.

Fig. 22. Illustration for the explanation of property 5

Property 6

The asymptotes of the ellipse intersect on a circle passing through four foci of hyperbolas (Fig. 23, a). The asymptotes of both hyperbolas also intersect on this circle (Fig. 23, b, c)

Fig. 23 a. Illustration for explaining property 6 with respect to an ellipse

Fig. 23 b Illustration for the explanation of property 6 with respect to blue hyperbola

Fig. 23, c. Illustration for the explanation of property 6 with respect to red hyperbola

Property 7

The circles generated from imaginary and from real foci are orthogonal.

Fig. 24. Illustration for the explanation of property 7

Property 8

Lines drawn through the center of the ellipse and the intersection points of unpaired focal circles intersect, including tangents and directrixes of adjacent hyperbolas.

Fig. 25. Explanation to property 8

Property 9

The straight lines drawn from the center of the ellipse to the intersection points of the directrixes of the hyperbolas intersect, including at the intersection points of the diametrical circles constructed on these points.

Fig. 26. Explanation to property 9

Property 10

The circle drawn through the focal points of adjacent conjugate hyperbolas also passes through one of the intersection points of the diametrical circles constructed on the intersection points of the directrixes of conics

Fig. 27. Illustration for property 10

Property 11

A line tangent to an ellipse at the point of intersection with the directrixes of adjacent hyperbolas passes through the foci of these hyperbolas.

Fig. 28. Illustration for property 11

Property 12

Lines drawn from the actual focus of one of the hyperbolas touch the adjacent hyperbola at points lying on the circle passing through the real foci of these hyperbolas. These points also lie on the directrix formed from the pair focus of the initial hyperbola.

Fig. 29. Illustration for property 12

Property 13

A circle centered at one of the foci of the hyperbola drawn through the points of tangency of the lines dropped from the pair focus to the adjacent focal circle touches the adjacent conjugate hyperbola at the intersection points of the adjacent hyperbola with the circle drawn through the real foci of the conjugate hyperbolas.

Fig. 30. Illustration for property 13