TRANSFORMATION OF SECOND-ORDER CURVES AND THEIR CHARACTERISTICS IN RELATION TO THE ELLIPSE

ABSTRACT

This article extends the notion of circle inversion to a broader context. After examining various methods for point inversion and assessing the outcomes when applying inversion to both straight lines and circles, this paper presents a novel approach for inverting arbitrarily chosen geometric shapes. By employing an affine transformation originating from the circle, an analytical expression for inversion concerning ellipses is derived. Additionally, the article establishes a formula for elliptical inversion using elliptic numbers. Furthermore, as a result of employing inversion, certain fourth-order curves emerge. The article also includes complex plane graphs illustrating these curves.

АННОТАЦИЯ

В данной статье расширяется понятие инверсии окружности в более широком контексте. После изучения различных методов инверсии точек и анализа результатов применения инверсии к прямым линиям и окружностям представлен новый подход к инверсии произвольно выбранных геометрических фигур. Посредством аффинного преобразования, исходящего из окружности, выведено аналитическое выражение для инверсии относительно эллипсов. Кроме того, установлена формула эллиптической инверсии с использованием эллиптических чисел. В результате применения инверсии также появляются определённые кривые четвертого порядка. В статье представлены графики на комплексной плоскости, иллюстрирующие эти кривые.

Keywords: second-order curve transformation, elliptical inversion, affine transformation, complex plane analysis.

Ключевые слова: преобразование кривых второго порядка, эллиптическая инверсия, аффинное преобразование, анализ на комплексной плоскости.

Introduction. In this article, we study the inversion of second-order curves as well as some of its associated properties. Elliptic inversion is a generalization of classical inversion, which has numerous properties and applications [1–5].

The structure of this work is as follows:

• Sections 1 and 2 define inversion with respect to a circle and an ellipse. These sections examine the Cartesian coordinates of elliptic points and also present the inversion formulas for an ellipse in a Cartesian system.
• Section 3 describes the equations of a line and a circle in the elliptic plane.
• Section 4 discusses inversion with respect to a circle in an oblique coordinate system and outlines the fundamental properties of elliptic inversion.

The purpose of this research is to extend the classical concept of inversion, traditionally applied to circles, to a broader context involving ellipses and other second-order curves. By examining various methods of point inversion and analyzing their effects on geometric shapes, this study aims to derive analytical expressions for elliptical inversion using affine transformations. Furthermore, the research seeks to explore the emergence of fourth-order curves resulting from these transformations and to establish a comprehensive framework for understanding the geometric relationships that arise from such operations. Ultimately, this work aspires to contribute valuable insights into geometric transformations, enhancing both theoretical knowledge and practical applications in fields such as mathematics, physics, and engineering.

Materials and methods

This section outlines the methodologies employed in the study to provide a structured understanding of geometric transformations through the process of inversion. The methods encompass deriving analytical expressions for both circular and elliptical inversions, utilizing complex numbers and affine transformations. By establishing a clear mathematical framework, this section aims to comprehensively analyze how second-order curves behave under various inversion scenarios, thereby enhancing the overall understanding of their geometric properties.

1. Inversion with respect to a circle in the Cartesian coordinate system

Let's establish a rectangular coordinate system  on the plane. Then for each complex number  represented in algebraic form  it is possible to uniquely associate a point on the plane, where  is the complex coordinate of point .

Let's write down the formula for inversion with respect to a circle in complex numbers. Suppose we have a circle with a radius of r and its center at the point :

 From the definition of inversion, we know that if the center of the inversion circle is located at point  and its radius is r then the inversion formula takes the following form (Fig. 1):

where  is the inversion power.

Inversion (2) provides an example of a nonlinear transformation. Indeed, if the center of inversion has coordinates , then

These formulas represent an analytical expression for inversion with respect to a circle.

2. Inversion with respect to an ellipse in the Cartesian coordinate system (affine transformation method)

Starting from the circle (1) through an affine transformation

we obtain an ellipse represented by the equation [6]:

where , , , , , , . Here  represents the coordinates of the ellipse’s center.

Figure 1. Inversion with respect to a circle

A reminder: If the general equation of a second-order curve

defines an ellipse, then , , where , ,  are the invariants of the curve (5).

Question: How is the transition from equation (5) to (1) carried out?

Answer: To achieve this, let’s consider the following affine transformation

In this case, the Jacobian of the affine transformation  is nonzero. Now, to obtain a similar formula for inversion with respect to an ellipse centered at the point , we will use the affine formulas (2). Then, formulas (3) provide the following analytical expressions

which define inversion with respect to an ellipse centered at the point . In the particular case where the origin coincides with the center these formulas simplify. For example, when , equation (1) transforms into the equation

,

then the inversion formulas (6) can be represented as:

, .

In turn, the inversion formulas (7) yield

, .

Here  and , . Then in the rectangular Cartesian coordinate system   inversion with respect to the ellipse is expressed as

, .

3. Equation of lines and circles in the elliptical plane

In the Euclidean plane, we introduce an oblique coordinate  system using control parameters  and  (Fig. 2). Then for each elliptical number  represented in algebraic form  ,  , , a unique point  on the plane can be associated.

Figure 2. The oblique coordinate system

Note 1: In the oblique coordinate system (Fig. 2) the direction of the imaginary axis (i.e., the direction of the basis vector , , ) relative to the Cartesian coordinate system  is determined by the condition

Note 2: The direction of the radius vector  relative to the Cartesian coordinate system  is determined by the condition

Let's consider a line passing through the point , where ,  are the real coordinates of point  in the oblique coordinate system, parallel to the vector , . Then the parametric equations of this line will have the form

, .

If we multiply the second equation by p and add it to the first equation, we obtain the parametric equation of the line in elliptical numbers as follows:

The general equation of a line on the plane is given by . Substituting

,

into this equation, we find

,

By introducing the notation  we finally obtain the equation of the line in elliptical form as

where , as .

The general equation of a circle in the oblique coordinate system  is expressed as

Expanding the brackets, we can see that equation (12) is indeed equation (5):

.

Substituting

,  and

into this equation, we get

where , ,  as .

Thus, in the oblique coordinate system, the equations of a line and a circle can be unified into a single equation

where  for the line and  for the circle. Moreover, if the circle is defined by equation (13), it is easy to calculate its center and radius

4. Inversion with respect to a circle in the Oblique coordinate system

In the oblique coordinate system , the equation  defines a circle with its center at the point  and radius , where , , , . Inversion with respect to this circle in the oblique coordinate system is given by the formula

where .

We know that the equation  in the original Cartesian coordinate system  defines an ellipse. Therefore, formula (16) defines inversion with respect to an ellipse in the Cartesian coordinate system :

.

Thus, if we consider the Cartesian coordinate system , inversion with respect to the ellipse  in coordinate form is defined as

Consider a specific case

1.  Let , . Then the oblique coordinate system becomes Cartesian, as , , , . Consequently, the formulas

,

define an inversion with respect to a circle.

2. Let the ellipse in the Cartesian coordinate system  be defined by its canonical equation

Equation (18) can be compared with the general equation of an ellipse . After the comparison, we obtain: , , , , , , where. Then the inversion formulas (17) with respect to an arbitrary ellipse give the analytical expressions for the inversion with respect to the canonical equation of an ellipse:

In the oblique coordinate system, let us determine into which sets of points lines and circles are transformed under inversion. In the case where the origin coincides with the center of inversion, the inversion formulas are simplified and allow us to prove the following properties:

1. The image of a line passing through the center of inversion coincides with this line;
2. The image of a line that does not pass through the center of inversion is a circle that passes through this center;
3. The image of a circle passing through the center of inversion is a line that does not pass through this center.

Problem 1. In the oblique coordinate system , a line  is given, passing through the center of inversion. Find the image of this line under inversion with respect to the circle , where , .

Solution. The general equation of a line in complex conjugate coordinates is expressed as (11): , i.e.,

where , , . To find the image of the line under inversion, we take the inversion equation  and substitute it into equation (20):

.

If we remove the primes, the image of the line satisfies equation (20). Thus, under inversion, the line passing through the center of inversion remains unchanged (Fig. 3).

In the Cartesian coordinate system , we have an ellipse  and a line  passing through the center of the ellipse. In this case, the inverse image of the line with respect to the center of the ellipse also remains unchanged (Fig. 3).

, .

Figure 3. The inverse image of the line with respect to the center of the ellipse

Problem 2. In the oblique coordinate system , consider a line that does not pass through the center of inversion. Find the image of the line  under inversion with respect to the circle , where , .

Solution. . Substituting  in this equation, we obtain the image of the line under inversion . The resulting equation is that of a circle. In this equation, the constant term is equal to zero. Therefore, the circle passes through the center of inversion.

In the Cartesian coordinate system , we have an ellipse  and a line  that does not pass through the center of the ellipse. In this case, the inverse image of the line with respect to the center of the ellipse transforms into an ellipse  that passes through the center of inversion (Fig. 4).

Figure 4. The inverse image of the line that does not pass through the center of inversion

Problem 3. In the oblique coordinate system , consider a circle passing through the center of the inversion . Find the image of this circle under inversion with respect to a circle , where , .

Solution. After inversion , the circle transforms into a line  that does not pass through the center of inversion.

Indeed, in the Cartesian coordinate system , we have an ellipse  that passes through the center of inversion. Then the inverse image of the ellipse with respect to the center of the original ellipse becomes a line that does not pass through the center of inversion (Fig. 5).

Figure 5. The inverse image of an ellipse that passes through the center of inversion

Results and discussion

The study presents significant findings regarding the transformation of second-order curves through inversion, particularly focusing on the extension of classical inversion from circles to ellipses. The results are categorized into key areas of discovery, which are discussed in detail below. One of the primary outcomes of this research is the successful derivation of analytical expressions for inversion with respect to ellipses. By employing affine transformations, the authors established a mathematical framework that extends traditional circular inversion concepts. The derived formulas for elliptical inversion demonstrate how points transform when subjected to this geometric operation, revealing intricate relationships between various geometric entities. This advancement not only enriches the theoretical understanding of geometric transformations but also provides practical tools for further applications in mathematics and related fields. The application of inversion techniques resulted in the emergence of specific fourth-order curves, which were illustrated through complex plane graphs. These curves highlight the complexity and richness of geometric transformations that occur when moving from simple circular inversions to more complex elliptical scenarios. The identification of these fourth-order curves emphasizes the potential for discovering new geometric properties and relationships that can arise from such transformations. The research also successfully unifies the treatment of lines and circles under the framework of inversion. The findings indicate that lines passing through the center of inversion remain unchanged, while lines that do not pass through this center transform into circles that do intersect at the center. Conversely, circles passing through the center are transformed into lines that do not intersect it. This duality illustrates a profound symmetry in geometric transformations and enhances our understanding of how different shapes interact under inversion. The inclusion of complex plane graphs serves as a vital tool for visualizing the results. These representations not only substantiate the theoretical findings but also facilitate a clearer interpretation of how various geometric shapes behave under inversion. The visual aids enhance comprehension and allow for a more intuitive grasp of the underlying mathematical principles. The implications of this study are far-reaching, suggesting numerous avenues for future exploration. The methodologies developed can be applied to investigate other geometric transformations, potentially leading to new discoveries in both pure and applied mathematics. Furthermore, the insights gained may have practical applications in fields such as physics, computer graphics, and engineering, where understanding geometric relationships is crucial.

Conclusion In conclusion, the transformation of second-order curves, particularly ellipses, through inversion processes offers a deepened understanding of geometric relationships in both Cartesian and oblique coordinate systems. By extending the concept of inversion from circles to ellipses, this study provides analytical expressions for such transformations and demonstrates the emergence of complex fourth-order curves. The use of affine transformations allowed the derivation of explicit inversion formulas, offering a broader framework for geometric analysis. The results not only unify the treatment of lines and circles under inversion but also highlight the unique properties of elliptical transformations, such as the preservation or transformation of lines into circles depending on their relationship with the center of inversion. This work paves the way for further exploration of geometric transformations and their applications in various fields, including physics, computer graphics, and engineering.

References:

1. Inversion Theory and Confrontal Mapping, part of the Student Mathematical Library (Volume 9) / D. Blair – American Mathematical Society, 2000. – 117 pp.
2. Excursions in Geometry / S. Ogilvy – Dover Publications, 1991. – 192 pp.
3. Geometry: A Comprehensive Course / D. Pedoe – Dover Publications, 1988. – 464 pp.
4. Geometricheskie svoystva krivykh vtorogo poryadka [Geometric Properties of Second-Order Curves] / A.V. Akopyan, A.A. Zaslavsky – Мoscow; MTsNMO, 2007. – 136 pp. [In Russian].
5. Ruinskiy A. Inversnye preobrazovaniya giperboly [Inverse Transformations of Hyperbolas] // Matematicheskoe prosveshchenie, Series 3, 4 – 2000. – P. 120-126. [In Russian].
6. Sagindykov B.Zh., Jatykov T.E. Geometricheskaya interpretatsiya ellipticheskogo chisla [Geometric interpretation of elliptical numbers] // Dostizheniya vuzovskoy nauki 2021: sbornik statey XVII Mezhdunarodnogo nauchno-issledovatelskogo konkursa – Penza: MTsNS «Nauka i Prosveshchenie». – 2021. – P.12-17. [In Russian].