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The insufficient use of quadratic transformations in applied geometry is explained by the fact that the methods of quadratic transformations are not developed much, although dozens of works by leading experts in applied geometry are devoted to the study of this problem, the development of graphic models, and their application in applied geometry. The research is devoted to the development of the theory of definition of biquadratic transformations of the plane. The essence of the proposed method for modeling biquadratic transformations of the plane, generated by a binary mapping of two surfaces of the second order, facilitation and solution of complex problems of applied geometries. And also, by means of graphical models of biquadratic transformations of the plane to facilitate the construction of curves of the second and fourfold orders. Considering a combination of non-linear surfaces of the second order, obtain subgroups of biquadratic transformations of the plane. The developed algorithm will make it possible to determine mathematical models of canonical biquadratic transformations of the plane, which is necessary for their practical application.
quadratic transformations, Cremona transformations, binary surface models, non-linear surfaces
In applied geometry, the method of quadratic transformations is effective in solving positional problems related to curves or surfaces. Quadratic transformations are very important because they have great practical significance in the engineering. Using it allows you to halve the order of the projected curve or surface, which greatly simplifies the solution of various positional problems [1, 2]. Multivalued dot geometric transformation can be one – two-digit, one – digit, etc. From them one – two-point conversion is rather well studied.
The work by Öztürk and Özdemir [3] provides a more complete overview of the main issues of the classical theory of nonlinear birational transformations of the plane of three-dimensional space. Dantas and Pan [4] considered Cremona quadratic transformations – the basics of the classical theory of nonlinear birational (cremonic) transformations of the plane of three-dimensional space. In the above work [5], the conditions of transformation of the second-order surface and the plane were studied, and the properties of transformations were studied: for example, one and two-digit quadratic transformation having a bundle of direct carriers of the corresponding points. Öztürk and Özdemir [3] in their work gives a more complete survey of the basic questions of the classical theory of nonlinear birational transformations of the plane and three-dimensional space. This work is of great value because it contains the resolution of singularities of curves and surfaces. The author reviews previously published materials on Cremona transformations' construction and use. Farouki et al. [6] proposed a method for obtaining special Cremona transformations of three-dimensional space. The mapping described in this study is constructive in nature, and it generates not only a model of projective spatial geometry on a plane, but allows us to practically attribute their images to specific spatial objects in the sense of descriptive geometry. This essential feature of the mapping, which has an independent theoretical significance, makes it valuable from the point of view of applications to non-elementary problems of descriptive geometry in Euclidean space. Research by Farouki et al. [6] is devoted to the study of two – two-valued correspondence $\mathrm{T}_{2-2}$. This correspondence consists in constructing points $A_2^{\prime}$ and $\mathrm{A}_2^{\prime \prime}$ from point $A_1$, which are projections on the frontal plane of projection $\Pi_2$ of intersection points perpendicular to the horizontal plane of projection $\Pi_1$.
In a rectangular system of two planes $\Pi_1$ and $\Pi_2$, the projection of a second-order surface was taken – a second-order curve can be specified by its traces. The curve of the second order $b_1^2$ in the horizontal plane of the projections $\Pi_1$ is given by the traces of the projection plane Г, perpendicular to the plane of the projections $\Pi_1$. It is also given by an intersecting curve of the second order along a curve and a point belonging to the section of the curve of the second order, conjugate to the direction perpendicular to the plane of the projection $\Pi_1$ [6, 7].
The plane α is drawn through point $A_1$. The trace of the plane α on the horizontal projection plane intersects the contours of the curves $a_1^2$ at points $\mathrm{B}_1$ and $\mathrm{C}_1$, and the curve $b_1^2$ at points $D_1$ and $L_1$. Further, the points $D_2$ and $L_2$ belonging to the projection axis x are constructed. The rays drawn through points $B_1$ and $C_1$ intersect the curve $a_2^2$ on the frontal projection plane at points $B_2$ and $\mathrm{C}_2$. Through these points $D_2$, $L_2$ and $B_2$, $C_2$, the contours of the second-order curve $b_2^2$ on the frontal projection plane were drawn. A straight line drawn through point $A_1$ on the projection axis x intersects curve $b_2^2$ at points $\mathrm{A}_2^{\prime}$ and $\mathrm{A}_2^{\prime \prime}$. This construction made it possible to solve the problem of a given point belonging to a surface of the second order. If point $A_1$ describes a straight line, then points $A_2^{\prime}$ and $A_2^{\prime \prime}$ describe a curve of the second order and vice versa, then the correspondence $\mathrm{T}_{2-2}$ is quadratic. If point $\mathrm{A}_1$ describes a curve of the second order, then points $A_2^{\prime}$ and $A_2^{\prime \prime}$ describe two curves of the second order, then the correspondence $\mathrm{T}_{2-2}$ is biquadratic.
From the analysis and research of scientific works on geometry, it follows that quadratic transformations have been studied sufficiently and have found practical applications in science and technology, but little attention has been paid to the study of biquadratic transformations of the plane [8, 9]. The authors believe that this gap prevents the full use of biquadratic transformations of the plane in construction. Therefore, this article is devoted to the development of the theory of determining biquadratic transformations of the plane. As a result of the work, a scientific review of the history and ways of development of descriptive geometry in the Republic of Kazakhstan was conducted. In addition, the authors developed theoretical foundations for modeling biquadratic transformations of the plane, which allowed us to establish new patterns for obtaining four-four-digit correspondences between two non-displaced planes. The article consists of Introduction, Theoretical Overview, Materials and Methods, Results and Discussion and Conclusions. The degree of study of the topic, the importance of this research is presented in the Introduction. Theoretical Overview contains the main sources and theories on the selected topic. Results and Discussion comments and illustrates the obtained data. Conclusions contain a concise summary based on the results of the study.
Alberich-Carramiñana et al. [10] considered a theory for constructing geometric models of quadratic Cremona transformations of a plane and three-dimensional space, as applied to solving problems of applied geometry. The work of Martín-Pastor [2] is devoted to proving a theorem that provides two-two-digit quadratic correspondences between points of combined fields. It describes geometric transformation tools that allow you to move from complex geometric shapes to simple geometric shapes that are described using a smaller number of parameters.
A graphical algorithm of two-two-digit quadratic correspondence Martín-Pastor [2] in which, depending on the type and relative position of the fundamental curves k^{2}, m^{2}, as well as the position of the poles S=S^{/}, you can get a number of private algorithms for solving various applied problems. The polaritet is set on an irregular line, the double points of which are the points of intersection of the fundamental curve with the irregular line. The types of fundamental curves depend on the type of involutions. If the involution is infinite and has no double points, then the fundamental curve is an ellipse. If the involution is infinite and has two double points, then the fundamental curve is a hyperbola, in the special case, a pair of non-matching lines. If the involution is infinite and has one double point, then the fundamental curve is a parabola [2].
In the work of Baidabekov et al. [11] contains studies of the theory of central Cremona transformations of a plane and three-dimensional space, and also deals with the study of the relationship between the methods of descriptive and algebraic geometry in the design of surfaces. The presence of the relationship between the methods of two images and birational transformations in modeling and research of surfaces should contribute to the rational use of their advantages [12, 13]. It should be noted that the object of the research is the presentation of the theory of Cremona transformations of the plane, where much attention is paid to constructive issues of central transformations in solving problems of an applied nature. He also developed the theory of central involutional Cremona transformations of space.
Received by Zhang et al. [14] a constructive method for studying transformations of three-dimensional space, stratified in a bundle of planes into central quadratic involutions, was later used to construct the shape of surfaces such as pipelines of variable cross-section. Recommendations are developed for the transformation apparatus for quadratic involutions, as well as for the study of graphical and analytical methods of the properties of some surfaces of dependent sections of the fourth order, bearing the frame of conics.
In the work of Corcoran and Jones [15] by means of a double space-to-plane mapping in the classical method of two images, a cubic involutional transformation $J_{3-3}$ was obtained, which splits in a bundle of planes into central quadratic involutions $J_2$, and possible special cases of specifying a cubic involution $J_{3-3}$ in relation to the design of technical forms of surfaces were studied and a method of approximation was developed surfaces based on the properties of the involution $J_{3-3}$.
In the quadratic transformation Martín-Pastor [2] (1), the (1-2)-valued transformations given by the equations were considered:
$\left\{\begin{array}{l}x=x^{\prime} \\ y=\pm \sqrt{\left(y^{\prime}\right)^2-\left(x^{\prime}\right)^2}\end{array}\right.$ (1)
where, x and y are the Cartesian coordinates of the point – the image; $x^{\prime}$ and $y^{\prime}$ are the Cartesian coordinates of the preimage point.
This correspondence is mutually one – two-valued and is called a quadratic transformation.
With the help of a quadratic transformation, methods for constructing an ellipse, a hyperbola, as well as methods for solving positional problems were obtained, which made it possible to open a new direction in the study and application of geometric transformations [16]. Quadratic transformations do not have such broad capabilities, in the sense of deformability, which are inherent in topological transformations, therefore, using quadratic transformations, only some regular curves, in particular, conical sections, can be transformed into a straight or homothetic curve.
In the work of Liu et al. [17] analytical methods of geometric modeling and solving inverse problems of birational and rational transformations of the n-th order of the plane generated by immersion in the model of three-dimensional spatial transformation of algebraic surfaces of monoidal and General types are investigated and developed. A method is proposed for constructing instantaneous Cremona transformations of the n-th order of three-dimensional space using specified guide curves, the number of which can be equal to n(n+3)/2+1, where n is the order of the Cremona transformation under consideration.
Thus, in applied geometry, the theory of quadratic transformations is developed quite fully, so when creating theoretical positions, we take as the basis of the research being conducted. From the analysis and research of scientific works on geometry, it follows that quadratic transformations have been studied sufficiently and have found practical applications in science and technology, but little attention has been paid to the study of biquadratic transformations of the plane [17].
The essence of the proposed method for modeling biquadratic transformations of the plane generated by binary mapping of two second-order surfaces is as follows.
In Euclidean three-dimensional space Е_{3}, two intersecting algebraic surfaces of the second order Ф_{1}^{0} and Ф_{2}^{0} are given (2, 3), whose equations have the form:
Ф_{1}^{0 }(Х_{1}, Х_{2}, Х_{3}) (2)
Ф_{2}^{0 }(Х_{1}, Х_{2}, Х_{3}) (3)
where, Х_{1}, Х_{2}, Х_{3 }are Cartesian coordinates; Ф_{1}^{0 }and Ф_{2}^{0 }are second-order continuous polynomials.
Two general position projection planes П_{1} and П_{1}^{/ }are also specified. We also draw a projecting beam S_{1}, which intersects the specified surfaces Ф_{1}^{0} and Ф_{2}^{0}, respectively, at points А_{1}^{0} and А_{2}^{0}, А_{3}^{0} and А_{4}^{0}. The surface of the second order Ф_{1}^{0} is rotated around the axis of ОХ_{2} by 90^{0 }so that the positive direction of the axis of ОХ_{1} coincides with the negative direction of the axis of ОХ_{3 }following Figure 1.
Figure 1. Projection of points А_{1}^{01} and А_{2}^{01} on the plane П_{1}^{/} = П_{1}
In other words, the second-order surface Ф_{1}^{0} undergoes a transformation q_{1} (rotation around the ordinate axis at an angle of 90^{0}), whose matrix is given by the equation:
$\left(\begin{array}{l}X_1^{\prime} \\ X_2^{\prime} \\ X_3^{\prime}\end{array}\right)=\left(\begin{array}{ccc}0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0\end{array}\right)\left(\begin{array}{l}X_1 \\ X_2 \\ X_3\end{array}\right)$ (4)
Figure 2. Projection of points А_{3}^{01 }and А_{4}^{01} on the plane П_{1}^{/} = П_{1}
Thus, we get a new position Ф_{1}^{01} of the second-order surface Ф_{1}^{0} and points А_{1}^{01} and А_{2}^{01}, which correspond to points А_{1}^{0} and А_{2}^{0}. The points А_{1}^{01} and А_{2}^{01 }are projected by the projecting beam S_{2} on the plane П_{2}, we get the points А_{1} and А_{2} following Figure 1. Let's consider an example of modeling a biquadratic transformation of the plane when the first surface Ф_{1}^{0} is a double-sided hyperboloid with a real axis ОХ_{3}, and the second surface Ф_{2}^{0} is a double-sided hyperboloid with a real axis ОХ_{1} following Figure 2.
According to formula (4), the surface Ф_{1}^{0} and the surface Ф_{2}^{0} are transformed (5):
$\left(\begin{array}{l}X_1^{\prime} \\ X_2^{\prime} \\ X_3^{\prime}\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{array}\right)\left(\begin{array}{l}X_1 \\ X_2 \\ X_3\end{array}\right)$ (5)
then it is orthogonally displayed on the projection plane П_{1}^{/} in accordance with Figure 1.
As a result of the above method, each point of the П_{1} plane is converted to four points of the П_{1}^{/} plane. Given a two-parametric set of points of the combined plane П_{1}^{/}=П_{1}, we obtain a biquadratic transformation of the plane. Similarly, it can be shown that in the opposite direction, each point of the П_{1}^{/} plane is transformed into four points of the П_{1} plane.
As a result of the sequential execution of the above design apparatus, each point of the П_{1} plane is converted into four points of the П_{1}^{/} plane. Thus, in the two parametric sets of points of the plane П_{1}^{/}=П_{1}, we get a biquadratic transformation of the plane L. Similarly, we can show that in the opposite direction, each point A^{/} of the plane П_{1}^{/} is transformed into four points of the plane П_{1}, this will be the inverse transformation of L^{/}. Given a two-parameter set of points of the combined plane П_{1}^{/}=П_{1}, we obtain a biquadratic transformation of the plane indicated by the letter L. Similarly, we can show that in the opposite direction, each point A^{/} of the plane П_{1}^{/} is transformed into four points of the plane П_{1}. This transformation is indicated by the letter L^{/} in accordance with Figure 3.
Figure 3. Getting points А_{1}^{/}, А_{2}^{/}, А_{3}^{/}, А_{4}^{/}
Figure 4. List of second-order non-linear surfaces
Using the above proposed spatial design scheme, various types of canonical biquadratic transformations of L, L^{/} plane are obtained, which are considered below. To obtain biquadratic transformations of the plane, two second-order surfaces are mapped to the plane. In this case, we consider cases where combinations of non-linear surfaces of the second order as a result of the implementation of these cases have received subgroups of biquadratic transformations of the plane. To model a subgroup of biquadratic transformations of the plane, in the above spatial design scheme, we consider the case when the combination of binary mapped surfaces of the second order are non-linear surfaces of the 2nd order, such as a sphere and a double-walled hyperboloid.
Figure 5. Combinations of non-linear surfaces of the 2nd order that participate in the spatial design scheme for obtaining biquadratic transformations
Figure 4 contains a list of second-order non-linear surfaces that serve as paired elements of the spatial design scheme. From these four surfaces, twelve combinations of the displayed surfaces Ф_{1}^{0} and Ф_{2}^{0} have been created, which are shown in accordance with Figure 5.
Studies have shown that, in accordance with Figure 5, items with sequential numbers 1-4, 7, 9, 10, 12 they form degenerate (4-4)-valued correspondences of the plane, and points with ordinal numbers 5, 6, 8, 11 allowed us to obtain four types of canonical biquadratic transformations of the plane L and L^{/}in accordance with Figure 6.
Figure 6. Biquadratic plane transformations generated by binary mapping of two non-linear surfaces
For the first time, the analysis of the research of scientists showed that there is a significant contribution of them to the development of theoretical provisions and methods of applied geometry. The data of this scientific review shows the history and ways of development of descriptive geometry in the Republic of Kazakhstan, and are also valuable for scientists, specialists in descriptive geometry.
The main result is the development of theoretical foundations for modeling biquadratic transformations of the plane. Regularities of generating biquadratic transformations of the plane are developed. The performed theoretical and applied research allows us to draw the following conclusions: the developed spatial design scheme for displaying two second-order surfaces allowed us to establish new patterns for obtaining four-four-digit correspondences between two non-displaced planes. Theoretical provisions for modeling canonical biquadratic transformations of the plane have been created. A method for obtaining biquadratic transformations of the plane generated by binary mapping of two second-order surfaces to a combined plane is developed. This method allowed us to obtain four types of canonical biquadratic transformations of the plane. The developed algorithm would make it possible to determine mathematical models of canonical biquadratic transformations of the plane, which is necessary for their practical application. The data obtained can be implemented in the surfaces design, as well as in other types of architectural design. The results of the research are recommended to be validated with experimental data for high-confident in the empirical formulations and findings. The future research direction can involve the development of the theory of biquadratic transformations of the plane.
Т_{2-2}. |
two-valued correspondence |
А_{1} |
point of the projection on the frontal plane |
В_{1} |
point of construction of rays 1 |
П_{1}. |
horizontal plane of projection |
П_{2} |
frontal plane of projection |
С_{1} |
point of intersection of rays 1 |
С_{2} |
point of intersection of rays 2 |
D_{1} |
point of intersection of curves and a plane 1 |
D_{2} |
constructing point 1 of the projection axis |
J_{2} |
central quadratic involution |
J_{3-3} |
cubic involutional transformation |
L_{1} |
point of intersection of curves and a plane 2 |
L_{2} |
constructing point 2 of the projection axis |
L_{3} |
biquadratic transformation 1 |
L_{4} |
biquadratic transformation 2 |
e |
constant number, parameter |
k^{2} |
fundamental curve |
L |
plane |
m^{2} |
fundamental curve |
n |
set of natural numbers |
O |
constructing point of the plane |
S |
pole 1 |
S^{/} |
pole 2 |
S_{1} |
beam |
x |
Cartesian coordinate of the point 1 |
y |
Cartesian coordinate of the point 2 |
А_{2} |
constructing point 1 |
А_{3} |
constructing point 2 |
А_{4} |
constructing point 3 |
Г |
projection plane |
ОХ_{1} |
axis 1 |
ОХ_{2} |
axis 2 |
ОХ_{3} |
axis 3 |
Ф_{1}^{0} |
second-order continuous polynomial 1 |
Ф_{2}^{0} |
second-order continuous polynomial 2 |
Х_{1} |
Cartesian coordinate 1 |
Х_{2} |
Cartesian coordinate 2 |
Х_{3} |
Cartesian coordinate 3 |
a |
curve |
b |
curve of the second order |
α |
plane |
[1] Cherunova, I., Tashpulatov, S., Davydova, Y. (2019). Geometric conditions of mathematical modeling of human heat exchange processes with the environment for CAD systems creating heat-shielding clothing. IOP Conference Series: Materials Science and Engineering, 680: 012039. https://doi.org/10.1088/1757-899X/680/1/012039
[2] Martín-Pastor, A. (2022). Polyhedral transformation based on confocal quadratic surface properties. Graphical speculations. Trends in Mathematics, 2: 35-51. https://doi.org/10.1007/978-3-030-99116-6_3
[3] Öztürk, İ., Özdemir, M. (2022). Affine transformations of hyperbolic number plane. Boletin de la Sociedad Matematica Mexicana, 28(3): 61. https://doi.org/10.1007/s40590-022-00455-2
[4] Dantas, D., Pan, I. (2018). Cremona transformations of P4 coming from complete intersection of quadrics. Geometriae Dedicata, 193(1): 73-88. https://doi.org/10.1007/s10711-017-0253-x
[5] Baidabekov, A., Baymukhanov, S., Kemelbekova, E. (2018). Graphical model of the biquadratic transformation. In IGGG 2018 – Proceedings of the 18th International Conference on Geometry and Graphics, Milan, Italy, pp. 147-160. https://doi.org/10.1007/978-3-319-95588-9_11
[6] Farouki, R.T., Knez, M., Vitrih, V., Žagar, E. (2021). Planar projections of spatial Pythagorean-hodograph curves. Computer Aided Geometric Design, 91: 102049. https://doi.org/10.1016/j.cagd.2021.102049
[7] Bartkov, M. (2021). Graal as a multilingual platform. Innovative Solution in Modern Science, 3(47): 141-151. https://doi.org/10.26886/2414-634X.3(47)2021.9
[8] Baidabekov, A., Kemebelkova, E. (2019). Areas for the existence of biquadratic transformations. Journal of Physics: Conference Series, 1260: 072002. https://doi.org/10.1088/1742-6596/1260/7/072002
[9] Koval, Y.M., Kutsova, V.Z., Kotova, T.V., Kovzel, M.A. (2021). Forecast and control of structure and properties of ultra-low-carbon steels. Metallofizika i Noveishie Tekhnologii, 43(6): 753-768. https://doi.org/10.15407/mfint.43.06.0753
[10] Alberich-Carramiñana, M., Ferragut, A., Llibre, J. (2021). Quadratic planar differential systems with algebraic limit cycles via quadratic plane Cremona maps. Advances in Mathematics, 389: 107924. https://doi.org/10.1016/j.aim.2021.107924
[11] Baidabekov, A., Baymukhanov, S., Kemelbekova, E. (2019). Graphical model of the biquadratic transformation. Advances in Intelligent Systems and Computing, 809: 147-160. https://doi.org/10.1007/978-3-319-95588-9_11
[12] Cozac, E.B. (2021). Complex algorithm for recovering binary images based on morphological methods of graphic dataprocessing. Interuniversity Collection of Scientific Works "Scientific Notes", 71: 35-42. https://doi.org/10.36910/6775.24153966.2021.71.5
[13] Denysenko, А.А. (2020). Deep learning for image classification in different color systems. International Journal of Applied and Fundamental Research, 8: 42-47. https://doi.org/10.17513/mjpfi.13114
[14] Zhang, S., Sun, L., Chen, Y., Bi, Y. (2020). Calibration method of three-dimensional plane array laser radar based on space-time transformation. Applied Optics, 59(28): 8595-8602. https://doi.org/10.1364/ao.400265
[15] Corcoran, P., Jones, C.B. (2018). Stability and statistical inferences in the space of topological spatial relationships. IEEE Access, 6: 18907-18919. https://doi.org/10.1109/ACCESS.2018.2817493
[16] Kozak, Y. (2021). Peculiarities of the development of algorithms for scheduling tasks within the framework of the concept of the edge computing. Computer-Integrated Technologies: Education, Science, Production, 43: 36-41.
[17] Liu, T.W., Bai, J.B., Lin, Q.H., Cong, Q. (2021). An analytical model for predicting compressive behaviour of composite helical Structures: Considering geometric nonlinearity effect. Composite Structures, 255: 112908. https://doi.org/10.1016/j.compstruct.2020.112908