A single constructive algorithm for constructing foci of second-order curves

Voloshinov Denis Vyacheslavovich Doctor of Technical Science 193232, Russia, g. Saint Petersburg, ul. Pr. Bol'shevikov, 22, of. korpus 1 denis.voloshinov@yandex.ru

The principles of determining the focal points of second-order curves are covered in scientific and pedagogical literature so widely and in detail that an attempt to find something new and meaningful in this matter may cause the reader deep surprise and bewilderment. Nevertheless, the article submitted to the readers' court calls for attention to seemingly well-known facts and well-established ideas from a slightly different point of view than it is customary to do in the mathematical literature. The reasoning is supposed to be carried out without using the analytical apparatus of mathematics based on the constructive and geometric properties of the studied images and their properties. These arguments, based on the apparatus of projective geometry, will reveal a number of contradictions in the currently existing definitions relating to second-order curves, and their elimination will provide an opportunity to develop a unified approach to the construction of some geometric images initiated by second-order curves and give them a general constructive justification.

As you know, affine geometry, which does not operate with the concept of infinity, distinguishes several types of second-order curves, among which in the future we will be interested in, in particular, the ellipse and hyperbola. From the point of view of projective geometry, second-order curves do not differ, as a result of which the algorithms for obtaining certain images associated with the concept of a conic section also do not differ. The more surprising is the fact that the issues of geometric justification of such images as focal points of conics in projective geometry have been left without due attention, and the known schemes for constructing these points are interpreted based on metric considerations and differ for ellipses and hyperbolas. This state of affairs cannot be called satisfactory, especially if incomplete and sometimes contradictory theoretical provisions are laid down as the basis for automation of geometric modeling procedures, since in practice this leads to a violation of the consistency and stability of these tools.

This is exactly the situation that has developed with the interpretation of second-order curves in the development of the Simplex system ^[1], designed to synthesize constructive geometric models not only using the apparatus of projective geometry, but also operating with imaginary images that inevitably arise in this geometry. Numerous experiments and analysis of the resulting geometric schemes carried out using this system ^[2-4] allowed us to conclude that some definitions related to the interpretation of second-order curves, which are the basis of geometric theory, are incorrect. In particular, the concept of the center of the second-order curve and the absence of the second main diameter of the ellipse is incorrectly interpreted. Rethinking this geometric phenomenon and adopting definitions in a new interpretation as a basis allows us to develop a unified approach to solving problems involving second-order curves and unify the functions of the geometric modeling system associated with these problems.

Usually, the center of a second-order curve is understood as a pole, in the polar transformation induced by this curve of an infinitely distant straight line, taken as a polar ^[5]. This definition is equally used to find the centers of non-degenerate curves of the second order: both ellipses and circles, and hyperbolas in affine interpretation. As you know, any collinear transformation defined in the plane translates a point to a point, a straight line to a straight line and a conic to a conic. At the same time, the incident property of the original objects and their images is preserved, while the metric properties of objects, in general, are not. In accordance with the definition of the center of the second-order curve used, again in the general case, in the collinear transformation, the center of the original curve does not pass into the center of the image curve.

Let's set ourselves the problem in the reverse formulation: let's say there are two conics a and b on the plane. It is required to choose such a collineation that would not only translate conic a into conic b, but also establish a correspondence between the main diameters of these conics. Let us first consider this problem based on the assumption that both conics are ellipses. Since the ellipses have two main diameters, it will not be difficult to find the intersection points of the corresponding diameters with the ellipses: P ₁, Q ₁ , R ₁ and S ₁ for the first ellipse and P ₂ , Q ₂ , R ₂ and S ₂. The collinear transformation provides not only the translation of the conic a into the conic b, but also the correspondence in the collineation of points and , i.e. . Such a transformation of a particular kind translates the center of one ellipse into the center of another (Fig. 1).

Fig. 1. A collinear transformation that translates an ellipse into another ellipse

 while maintaining the correspondence of the main diameters of the ellipses

Now we will perform a similar procedure, but with the only difference that we will establish a correspondence between the hyperbola a and the ellipse b. At this stage, the first difficulty arises: it is well known that the hyperbola has only one real principal diameter. The second additional diameter, usually called "imaginary", despite the fact that it passes through the center and is represented by a real straight line, intersects the hyperbola at two imaginary complex-conjugate points and an attempt to establish a collineation similar to the one discussed above will not translate conic a into conic b. However, it is easy to make up for the absence of a second real diameter in a hyperbola by abandoning affine representations of the problem being solved and moving on to the concepts of projective geometry. Let's make an assumption that the second diameter of the hyperbola is an infinitely distant line and check whether this assumption does not contradict any other properties of second-order curves. The first thing to note is that this diameter is real, and not "imaginary", which is usually attributed to hyperbole. The second and extremely important property for solving this problem is that the conic twice intersects an infinitely distant line at various infinitely distant points, which should take place on the diameter of the conic. Thus, having made this assumption, we obtain four points on the conic: two proper P ₁, Q ₁ from the only proper diameter of the hyperbola and two improper points of the plane R ₁^?, S ₁^? from the improper diameter of the conic. Let 's set the collineation . In this collineation, the hyperbola a will completely transform into the ellipse b, and . It should be noted that in this collineation, an improper point corresponds to the point, that is, the center of the ellipse b is not the center of the hyperbola a (Fig. 2). The apparent contradiction, however, turns out to be quite constructive.

Fig. 2. A collinear transformation that translates an ellipse into another ellipse

while maintaining the correspondence of the main diameters of the ellipses

We define the inverse collineation and draw through the point O ₂ the set of diameters {d ₂} of the curve b. By performing the transformation, we obtain a set of lines parallel to the proper diameter of the hyperbola a, which, due to their parallelism, intersect at a single point O ₁^? with the second improper diameter of this curve (Fig. 3). If we consider the result of the intersection of objects of this set with the "imaginary" diameter of the hyperbola, it is easy to make sure that this result is an infinite set of points on a straight line intersecting with the conic a at imaginary points. In this context, it is not necessary to talk about an imaginary diameter as an object that exhibits any properties of the diameter of the curve. But it is quite appropriate to assume that all the different lines of the set {d ₁} intersect at a single point O ₁^?, at which, among other things, the improper main diameter of the hyperbola intersects with these diameters. For this and for a number of other reasons, which will be given below, it is rational to consider this point O 1_{? as the center of the hyperbola} and abandon the definition stating that the center of the second-order curve is the result of the polar transformation of an infinitely distant line with respect to this conic ^[5].

Fig. 3. Correspondence of ellipse diameters and hyperbola diameters

in collineation transformation

Let's draw tangent lines through the intersection points of the main diameters of the ellipse with the ellipse itself and find the result of their transformation in collineation ^-1. As a result, we get four straight lines, two of which will be tangent to the conic a at points P ₁ and Q ₁, and the other two will become asymptotes approaching the hyperbola and connecting to it at infinitely distant points R ₁^? and S ₁^?. In this context, the last two lines can be considered tangent to the conics and do not distinguish between them and tangent lines in the usual sense (Fig. 4).

Fig. 4. Correspondence of tangents at the points of intersection of conics

with major diameters in collineation transformation

The performed transformation makes it possible to detect commonality in the schemes of constructing the focal points of an ellipse and a hyperbola and to reduce them into a single algorithm.

Let us first consider the procedure for constructing the focal points of an ellipse (Fig. 5). Let an ellipse b be given on the plane and the points P, Q, R, S of intersection of its main diameters f and g with itself be found. Let 's construct a point T as the result of the intersection of tangents and . Draw a circle u centered at point P and through point T. This circle will intersect the main diameter g at points F and F', which are the focal points of curve b on the g axis.

Fig. 5. Construction of the focal points of the ellipse

Acting similarly, we construct a circle v passing through the point T with the center at the point R. Finding the intersection of the circle v with the diameter f, we find two more foci F" and F"' of the conic b, but already on the f axis. Thus, we have obtained two pairs of focal points of the ellipse, each of which can be formed either by real or imaginary points, and both pairs cannot be simultaneously real or imaginary. Let's draw a circle j through the points F and F', taken as diametrical. It is easy to see that this circle is orthogonal to the circles u and v, as well as the main axes f and g of the conic b. The same will be true for the circle i, drawn through the points F" and F"', taken as diametrical. If one of the circles i or j is real, then the other is necessarily imaginary and vice versa, while both circles turn out to be concentric, and their center is a real point.

Let us now turn to the method of constructing hyperbola foci. With regard to the method of constructing focal points of this curve, there is a scheme similar to the scheme of constructing ellipse foci, but it has some differences that we have to eliminate as a result of analyzing these schemes. As in the case of an ellipse, it is necessary to find focal points on both main diameters of the hyperbola by constructing circles. But, since one of the diameters of the hyperbola is an infinitely distant straight line, the intersection points of any given circle with this straight line are fixed cyclic points of the plane, so the task of constructing them does not make much sense. It should also be noted that the cyclic points of the plane are imaginary. Therefore, we will build focal points only on the proper diameter of the conic a.

Let's construct a point T as the result of the intersection of tangents and a straight line n tangent to the conic a at an improper point S^?.Let's choose the center Z of the circle u, draw it through the point T. The focal points F and F' are defined as the intersection points of the circle u with the straight line f (Fig. 6).

Fig. 6. Construction of focal points of the hyperbola

It follows from the performed constructions that, in general, the algorithms for constructing the focal points of an ellipse and a hyperbola are the same. The difference lies only in the rule of choosing the position of the center Z of the circle u. To unify the algorithms, the unified choice of the position of the point Z can be interpreted as follows: the point Z must be on the tangent n coming from the intersection point of the main diameter with the curve itself, conjugate to the one on which the focus is currently being searched. In this case, the point T through which the circle passes should be considered as diametrical. The second diametric point T' is determined by the intersection of the tangent n with the second tangent m', the dual of m.As a result of this choice, the point Z will occupy the position of points P and R necessary for solving the problem when constructing the focal points of the ellipse.

As a result of the analysis of geometric schemes, a number of concepts of projective geometry were clarified, which made it possible to unify the solution of problems related to the construction of focal points of second-order curves. A unified algorithm for constructing all four foci of the second-order curve is presented. Thus, the foundation has been laid for expanding the application areas of geometric models to imaginary geometric images covered by the concept of a "second-order curve", and conducting research on the resulting geometric images and schemes.