Аналитическая проективная геометрия для компьютерной графики

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Мотивом к написанию данной работы послужила разработка авторами курса по компьютерной геометрии для студентов физико-математических специальностей. Под термином «компьютерная геометрия» здесь и далее понимаются математические основы машинной графики. Важно отдельно подчеркнуть, что разрабатываемый курс должен быть рассчитан на студентов второго года обучения и, следовательно, от них можно требовать лишь предварительное знание стандартного курса алгебры и математического анализа. Это накладывает определённые ограничения на излагаемый материал. При изучении тематической литературы было выяснено, что стандартом де факто в современной компьютерной графике стало использование проективного пространства и однородных координат. Однако авторы столкнулись с проблемой методологического характера - практически полным отсутствием подходящей учебной литераторы как на русском, так и на английском языках. Для представления собранной авторами информации по данному вопросу и была написана данная работа.

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1. Introduction Here are the main reasons that motivated us to write this paper. 1. Synthetic and analytical approaches to geometry Historically, there have been two approaches to the presentation of geometry in mathematics, synthetic and analytical. In the synthetic presentation of geometry, sets of geometric elements of various kinds are initially introduced, such as points, lines and planes. Then the relationship between them is defined, formulated in the form of axioms that correspond to visual geometric representations. This approach is used in a simplified form when presenting Euclidean geometry in a school geometry course, therefore it is intuitively understandable to most students. An alternative approach emerged later, with the development of algebra. Despite the fact that the name analytical geometry has been assigned to it, it would be more correct to call it linear-algebraic, “since linear algebra forms the basis and provides its own methods, not analysis” [1, p. 12]. The linear-algebraic approach is much more general, much more powerful, and therefore much simpler in describing complex structures than the synthetic approach. He received a particularly strong impetus for his development within the framework of the ideas formulated by Felix Klein in his Erlangen program [2]. It is quite natural for computer graphics algorithms to use a linear-algebraic approach, since it allows you to write down and use algebraic formulas to calculate the necessary quantities. Consequently, projective geometry in the framework of this subject should be presented in this style. Note that we are talking about the projective space model ℝP3. 2. Lack of educational literature The literature on projective geometry in Russian is extensive, not to mention English and other foreign languages. Most textbooks, where the presentation is conducted at a level accessible to undergraduates of physics and mathematics faculties, are essentially based on a synthetic approach [3- 7]. This technique is justified, since the task of these authors is to provide an understanding of the essence of projective geometry, and the analytical approach “requires more ink and less thought” [8, p. 89]. However, when presenting the basics of projective geometry in a computer geometry course, this approach is questionable because it is too far from the final software implementation. There are also a large number of monographs where projective geometry is presented in a linear- algebraic style, for example [9-11] and a list of sources in [1]. However, the style of presentation in them is rather abstract and most of them are textbooks for undergraduates, graduate students and researchers working in the framework of theoretical mathematics. Looking at textbooks on computer graphics, machine vision and robotics, then a different problem arises. In many textbooks, projective geometry begins and ends with an exposition of the concept of homogeneous coordinates, which are introduced exclusively in the context of projective transformations. For example, let’s list the sources with the pages: [12, pp. 101, 115][13, p. 18][14, pp. 192, 220][15, p. 146] [16, p. 85] [17, p. 20] [18, p. 176] [19, p. 211] [20, p. 438] [21, p. 56]. In all these books, homogeneous coordinates are introduced ad hoc and used to represent affine transformations as a linear transformation (3×3 matrices on the plane and 4×4 matrices in space). The representation of a straight line and a plane using homogeneous coordinates is not considered, and projective geometry is not applied to standard problems of analytical geometry. The list of textbooks in Russian concerning the mathematical foundations of computer graphics is extremely limited [12-14, 17, 19, 20, 22, 23]. None of these manuals consistently use projective geometry as a tool for solving computer geometry problems. Some information, mainly about homogeneous coordinates, is available in books [12-14, 17, 19, 24], however, they do not consistently describe how to represent straight lines and planes in a homogeneous form, and homogeneous coordinates are also used only to represent affine transformations as linear. As an exception, the textbook should be noted [25], which adopted a non-standard approach to the presentation of analytic geometry using Grassmann algebra using non-standard notation. We will also mention several sources on the theory of screws [26-28], in which, in particular, the moment of a sliding vector is introduced, which is directly related to the representation of straight lines in Plucker coordinates (in a homogeneous form). 76 Modeling and Simulation DCM&ACS. 2025, 33 (1), 74-102 The situation with books in English is only slightly better. You can specify the book [29], where the presentation is not limited to the introduction of homogeneous coordinates only for points, on the contrary, homogeneous coordinates for straight lines and planes are considered, as well as solutions to some standard problems [29, pp. 25, 65]. As a disadvantage, we note that all formulas are written mainly in component rather than vector form, and we also note the focus on the field of computer vision rather than computer graphics. These are different areas, despite their proximity. Of particular note is the extremely capacious but extremely informative book [30] by Eric Lengyel, which stands out in several ways. § The presentation is conducted at a good mathematical level, but with an emphasis on practical application. For almost every formula, an example of its implementation is given in the form of programs in C-like pseudocode. § Due attention is paid to the application of the principles of projective geometry in computer graphics problems. In the third chapter, the author provides an extremely useful table 3.1 with a summary of basic formulas using homogeneous coordinates for points, lines and planes. § The fourth chapter is devoted to Grassmann algebra, which is the basis of geometric algebra. The author gives a description of projective spaces in terms of
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Об авторах

М. Н. Геворкян

Российский университет дружбы народов

Email: gevorkyan-mn@rudn.ru
ORCID iD: 0000-0002-4834-4895
Scopus Author ID: 57190004380
ResearcherId: E-9214-2016

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация

А. В. Королькова

Российский университет дружбы народов

Email: korolkova-av@rudn.ru
ORCID iD: 0000-0001-7141-7610
Scopus Author ID: 36968057600
ResearcherId: I-3191-2013

Docent, Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация

Д. С. Кулябов

Российский университет дружбы народов; Объединённый институт ядерных исследований

Email: kulyabov-ds@rudn.ru
ORCID iD: 0000-0002-0877-7063
Scopus Author ID: 35194130800
ResearcherId: I-3183-2013

Professor, Doctor of Sciences in Physics and Mathematics, Professor. of the Department of Probability Theory and Cyber Security of RUDN University; Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация; ул. Жолио-Кюри, д. 6, Дубна, 141980, Российская Федерация

Л. А. Севастьянов

Российский университет дружбы народов; Объединённый институт ядерных исследований

Автор, ответственный за переписку.
Email: sevastianov-la@rudn.ru
ORCID iD: 0000-0002-1856-4643
Scopus Author ID: 8783969400
ResearcherId: B-8497-2016

Professor, Doctor of Sciences in Physics and Mathematics, Professor of Department of Computational Mathematics and Artificial Intelligence of RUDN University

ул. Миклухо-Маклая, д. 6, Москва, 117198, Российская Федерация; ул. Жолио-Кюри, д. 6, Дубна, 141980, Российская Федерация

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© Геворкян М.Н., Королькова А.В., Кулябов Д.С., Севастьянов Л.А., 2025

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