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1
Сальков and Nikolay Sal'kov. "Gaspard Monge’s Descriptive Geometry Course." Geometry & Graphics 1, no. 3 (December 3, 2013): 52–56. http://dx.doi.org/10.12737/2135.
Full textAbstract:
The structure of course of lectures, proposed by Gaspard
Monge in the late 18th century, is considered in detail. Monge
foresees 120 lectures for descriptive geometry. The main course,
without shadows and perspective, is divided into 5 sections, each
of which is devoted to a certain themes. In the first section the
projections deriving is described, and two main goals are given.
Here we face with the notion of geometric locus, elements of differential
and projective geometries. The second section considers
the tangent planes and normals. These elements’ use in painting
and building are demonstrated. The third section has been devoted
to the theory related to construction of curved surfaces intersections.
The solution of tasks has been considered in the fourth section.
The fifth section has been devoted to the curved lines. Gaspard
Monge proposes the developed course of descriptive geometry not
as a set of interesting geometric tasks, but as a solution of specific
application tasks inherent to the industry, building, art, and military
science.
2
Bellone, T., F. Fiermonte, and L. Mussio. "THE COMMON EVOLUTION OF GEOMETRY AND ARCHITECTURE FROM A GEODETIC POINT OF VIEW." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-5/W1 (May 16, 2017): 623–30. http://dx.doi.org/10.5194/isprs-archives-xlii-5-w1-623-2017.
Full textAbstract:
Throughout history the link between geometry and architecture has been strong and while architects have used mathematics to construct their buildings, geometry has always been the essential tool allowing them to choose spatial shapes which are aesthetically appropriate. Sometimes it is geometry which drives architectural choices, but at other times it is architectural innovation which facilitates the emergence of new ideas in geometry. <br><br> Among the best known types of geometry (Euclidean, projective, analytical, Topology, descriptive, fractal,…) those most frequently employed in architectural design are: <br> &ndash; Euclidean Geometry <br> &ndash; Projective Geometry <br> &ndash; The non-Euclidean geometries. <br><br> Entire architectural periods are linked to specific types of geometry. <br><br> Euclidean geometry, for example, was the basis for architectural styles from Antiquity through to the Romanesque period. Perspective and Projective geometry, for their part, were important from the Gothic period through the Renaissance and into the Baroque and Neo-classical eras, while non-Euclidean geometries characterize modern architecture.
3
ZHANG, R. B., and XIAO ZHANG. "PROJECTIVE MODULE DESCRIPTION OF EMBEDDED NONCOMMUTATIVE SPACES." Reviews in Mathematical Physics 22, no. 05 (June 2010): 507–31. http://dx.doi.org/10.1142/s0129055x10004028.
Full textAbstract:
An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi–Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.
4
Афонина, Елена, and Elena Afonina. "Technique of algorithmic approach when training in graphic disciplines." Bulletin of Bryansk state technical university 2014, no. 2 (June 30, 2014): 161–65. http://dx.doi.org/10.12737/23285.
Full textAbstract:
Advantages and shortcomings of algorithmic approach are considered at the solution of problems of descriptive geometry and engineering graphics. application of algorithmic approach when studying one of subjects of engineering graphics – "projective drawing" is analyzed. some recommendations about improvement of quality of training of specialists on the basis of experience of "descriptive geometry and graphics" chair of bgtu are provided
5
Ho, Pei-Ming. "Riemannian Geometry on Quantum Spaces." International Journal of Modern Physics A 12, no. 05 (February 20, 1997): 923–43. http://dx.doi.org/10.1142/s0217751x97000694.
Full textAbstract:
An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective space and the two-sheeted space.
6
Сальков and Nikolay Sal'kov. "Descriptive Geometry — the Base of Computer Graphics." Geometry & Graphics 4, no. 2 (June 18, 2016): 37–47. http://dx.doi.org/10.12737/19832.
Full textAbstract:
Everyone knows that descriptive geometry is a science,
but there is no definition of the computer graphics. If computer
graphics is announced as a science, you can continue to call descriptive
geometry outdated and to demand of its abolition. But if the computer graphics has nothing to do with the science, and if
it is only a tool to perform the procedures of descriptive geometry
and other branches of geometry, everyone who tried to discredit
descriptive geometry, will be in a difficult position: they will have
nothing more to say, because in it is unwise to replace science on
tool. It is known that engineering graphics so far does not have the
status of science. It is a discipline that fully applies the laws of
other geometrical sciences.
There are many questions such as: why are there claims that
descriptive geometry is 2D? Why do some specialists claim that
descriptive geometry is a projection on a single plane? They forgot
that descriptive geometry uses the method of two images or method
of two traces. But this method of two images is used everywhere!
On the first lecture we tell students: the projection is carried out
on two planes of projection, a point in space can be fixed only by
means of three coordinates and the point must have at least two
projections. It means, that descriptive geometry works with three
coordinates, in other words — in 3D. The image produced on the
display screen in the so-called 3D, is neither more nor less than
axonometric projection on the plane — appropriate section of
descriptive geometry.
In conclusion, the author offers to classify the computer graphics
as a tool for the experience of all branches of geometrical science,
but not as a free-standing science.
7
Сальков and Nikolay Sal'kov. "Descriptive Geometry As Theory of Images." Geometry & Graphics 4, no. 4 (December 19, 2016): 41–47. http://dx.doi.org/10.12737/22842.
Full textAbstract:
In April 2016 at the All-Russian seminar "Geometry
and Graphics", dedicated to the 125th anniversary of the birth of
Glazunov Yevgeny Alexandrovich on a replica of one of the
N.A. Salkov paper authors that descriptive geometry is the theory
of images, the speaker somehow ambiguously stated that it is not
quite so. How, in his opinion, was the case, he did not wish to
disclose. Moreover, some graphic disciplines teachers believe descriptive
geometry unworthy of study because some of them only
known causes. They do not understand that descriptive geometry
is the theory of images, and images are everywhere, right down to
the computer screen. There is no such area of the human labor
application where this or that image wouldn´t be applied. Images
are ubiquitous! That is why descriptive geometry as a theory of images should be studied not only in the artistic and technical high
educational institutions, but in all ones of Russia. In proposed
paper there are numerous examples of the classics (and not only
of them), expressed oneself on a subject of descriptive geometry.
All statements (of Gaspard Monge, academician N.F. Chetverukhin,
professors A.I. Dobryakov, E.A. Glazunov, V.O. Gordon, G.S.
Ivanov, S.M. Kolotov, V.A. Peklich, N.A. Rynin, S.A. Frolov, etc.)
are reduced to one and the same: a projection is an image, and
descriptive geometry is the theory of images. A diagram showing
that descriptive geometry’s interest area includes not only drawing,
but also thirteen other sections, has been presented.
8
Иванов, Gennadiy Ivanov, Дмитриева, and I. Dmitrieva. "The Nonlinear Forms in Engineering Geometry." Geometry & Graphics 5, no. 2 (July 4, 2017): 4–12. http://dx.doi.org/10.12737/article_5953f295744f77.58727642.
Full textAbstract:
The introduction of the methodology of a competence approach in teaching the courses of departments of engineering graphics, as shows the experience of a number of technical universities of Russia, occurs according to some template. If in 70–80 years the last century the Departments was the routing of the teaching subject with the indication providing, and provide discipline then currently are talking about interdisciplinary competences. In this case, interdisciplinary competence on engineering and computer graphics define the departments. The wording of the intrasubject competencies reduced to define the topics of descriptive geometry necessary for the study of the fundamentals of engineering graphics (geometrical and projection drawing, the cutting line and the intersection of simple surfaces). In the end, the important topics of descriptive geometry (basic concepts of the theory of curves and surfaces, sweep, tangent planes, axonometric etc.) that have practical value, are excluded from the syllabus. This greatly affects the quality overall geometric training of students of technical universities with based on the level of teaching geometry in high school. In our opinion, the way out of the situation is in consistent, purposeful transformation the descriptive geometry in the engineering geometry without radical distortions. In this regard, the present article is devoted to the presentation some issues in the theory of nonlinear forms in engineering geometry. The proposed approach will enhance the practical value of our discipline due to the expansion of cross-curricular competencies with related sections of higher mathematics, CAD, etc.
9
Иванов and G. Ivanov. "Previous History and Background of Transformation of the Descriptive Geometry in the Engineering Geometry." Geometry & Graphics 4, no. 2 (June 18, 2016): 29–36. http://dx.doi.org/10.12737/19830.
Full textAbstract:
In the past decade is widely discussed the problem of
geometric and graphic training of students of technical universities
in Russia. In 1980-2005 arose contradictions between traditional
and innovative methods of teaching for descriptive geometry and
engineering graphics. This marked the article Professor P.A. Tunakov,
in which descriptive geometry was carried to a dying science. This
radical statement in subsequent years was supported by V.A. Rukavishnikov
[15; 16] and A.L. Kheifets. An additional impetus to
discussions was given by the developers of the Federal state educational
standards of higher education (FSES), which declared the
competence approach to the process of learning and evaluation of
knowledge of graduates. Introduction in educational process of
computer graphics
and the appearance of technologies of 3D modeling
prompted some representatives of the departments of engineering
graphics towards the radical statements:
• descriptive geometry as a graphic discipline became "moribund",
"morally obsolete";
• it is necessary to refuse from the method of projection, as
"fundamentally important is a matter of conformity to the dimension
of the three-dimensional computer model and the
modeled object".
The article proves the incorrectness of these statements. History
and background of transformation of the descriptive geometry in
the engineering geometry are shown:
1) references to the dynamics of change subjects of presentations
at the Moscow seminars on descriptive geometry and on engineering
graphics during 1944–1965; the themes of dissertations
on the specialty 05.01.01 engineering geometry and computer
graphics (up to 1977 – applied geometry and engineering graphics);
2) the requirements of competence-based learning model to establish:
• intrasubject links (combination of synthetics and analytical
methods of problem solving);
• interdisciplinary connections by expanding the subject of
the discipline of the multidimensional shapes;
3) the incorrectness of opposing the "by the radicals" of 2D and
3D models, for they are complementary the types of modeling
single method of two images.
10
Sal'kov, N., and Nina Kadykova. "The Phenomenon of Descriptive Geometry Existence in Other Student Courses." Geometry & Graphics 8, no. 4 (March 4, 2021): 61–73. http://dx.doi.org/10.12737/2308-4898-2021-8-4-61-73.
Full textAbstract:
Among specialists prevails the primitive view, according to Prof. G.S. Ivanov, on descriptive geometry only as on a "grammar of a technical language", as it characterized V.I. Kurdyumov in the XIX Century. If in the century before last his definition was actual, although many contemporaries had a different opinion, then a century and a half later this definition became outdated, especially since have been revealed the close relationships of descriptive geometry with related sections: analytical, parametric, differential geometry, etc., and descriptive geometry became an applied mathematical science. In this paper it has been shown that an image is obtained as a result of display (projection). In this connection, according to prof. N.A. Sobolev, "All visual images – documentary, geometrographic, and creative ones – are formed on the projection principle". In other words, they belong, in essence, to descriptive geometry. Thus, all made by hand creative images – drawings, paintings, sculptures – can be attributed with great confidence to descriptive geometry as a theory of images. These creative images, of course, have a non-obvious projection origin, nevertheless, according to Prof. N.A. Sobolev, "They, including the most abstract fantasies, are essentially the projection ones". Further in the paper it has been shown which disciplines apply some or other of graphic models, and has been considered a number of drawings belonging to different textbooks, in which graphic models are present. Thus, clearly, and also referring to the authorities in the area of images and descriptive geometry, it has been proved that each of the mentioned textbooks has a direct or indirect connection with descriptive geometry, and descriptive geometry itself is present in all textbooks, at least, in the technical and medical ones.
11
Tessema, L. S., R. Jaeger, and U. Stilla. "A MATHEMATICAL SENSOR MODEL FOR INDOOR USE OF A MULTI-BEAM ROTATING 3D LIDAR." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-2/W16 (September 17, 2019): 227–34. http://dx.doi.org/10.5194/isprs-archives-xlii-2-w16-227-2019.
Full textAbstract:
<p><strong>Abstract.</strong> Our contribution presents a new perspective in the mathematical description of a rotating multi-beam LiDAR sensor, in a sense that we make use of projective geometry along with the “homogeneous general equation of the second degree” to parametrize scan lines. We describe the scan geometry of a typical multi-beam rotating 3D LiDAR by representing scan lines as pojective conics that represent a projective figure (a cone) in an embedding plane. This approach enables the parameterization of each scan line using a generic conic section equation. Most modeling approachs model spinning LiDAR sensors in terms of individual points sampled by a laser beam. On the contrary, we propose a model that provides a high-level geometric interpretation both for the environment and the laser scans. Possible application scenarios include exterior and interior calibration of multiple rotating multi-beam sensors, scan distortion correction and localization in planar maps.</p>
12
Adler, Allan. "Modular correspondences on X(11)." Proceedings of the Edinburgh Mathematical Society 35, no. 3 (October 1992): 427–35. http://dx.doi.org/10.1017/s001309150000571x.
Full textAbstract:
In this paper, we show how to give a geometric interpretation of the modular correspondence T3 on the modular curve X(11) of level 11 using projective geometry. We use Klein's theorem that X(11) is isomorphic to the nodal curve of the Hessian of the cubic threefold Λ defined by V2W + W2X + X2Y + Y2Z + Z2V = 0 in P4(C) and geometry which we learned from a paper of W. L. Edge. We show that the correspondence T3 is essentially the correspondence which associates to a point p of the curve X(11) the four points where the singular locus of the polar quadric of p with respect to Λ meets X(11). Our control of the geometry is good enough to enable us to compute the eigenvalues of T3 acting on the cohomology of X(11). This is the first example of an explicit geometric description of a modular correspondence without valence. The results of this article will be used in subsequent articles to associate two new abelian varieties to a cubic threefold, to desingularize the Hessian of a cubic threefold and to study self-conjugate polygons formed by the quadrisecants of the nodal curve of the Hessian.
13
Krasic, Sonja, and Miroslav Markovic. "Determination of the characteristic parameters in the special collinear space in the general case." Facta universitatis - series: Architecture and Civil Engineering 5, no. 1 (2007): 49–59. http://dx.doi.org/10.2298/fuace0701049k.
Full textAbstract:
The projective space consists of the finitely and infinitely distant elements. The special collinear spaces in the general case, are set with five pairs of biunivocally associated points, so the quadrangle in the first space obtained by the three principal and one penetration point of the remaining two through the plane of the first three identical or similar to the associated quadrangle obtained in the same way in the second space. In order to associate two special collinear spaces, it is necessary to determine the following characteristic parameters: vanishing planes, space axes (principal normal lines), foci (apexes of the associated identical bundles of straight lines) and directrix plane (associated identical fields of points). The paper is based on constructive processing of the special collinear spaces in the general case. The structural methods which are used are Descriptive Geometry (a pair of Mange's projections) and Projective geometry.
14
Волошинов, Denis Voloshinov, Соломонов, and K. Solomonov. "Constructive geometric modeling as graphic disciplines’ teaching prospect." Geometry & Graphics 1, no. 2 (July 25, 2013): 10–13. http://dx.doi.org/10.12737/778.
Full textAbstract:
The questions related to advisability of descriptive geometry teaching in high education institutions are considered. A comparison of descriptive and analytical geometry as well as a comparison of engineering graphics and computer one is given. The necessity of formation in the students’ mind а schematic projection principles is proved.
15
Howard, Benjamin, Christopher Manon, and John Millson. "The Toric Geometry of Triangulated Polygons in Euclidean Space." Canadian Journal of Mathematics 63, no. 4 (August 1, 2011): 878–937. http://dx.doi.org/10.4153/cjm-2011-021-0.
Full textAbstract:
Abstract Speyer and Sturmfels associated Gröbner toric degenerations of with each trivalent tree having n leaves. These degenerations induce toric degenerations of Mr, the space of n ordered, weighted (by r) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torus as “bendings of polygons”. We prove the conjecture of Foth and Hu that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida.
16
Серегин, V. Seregin, Иванов, G. Ivanov, Боровиков, and Ivan Borovikov. "Scientific and Methodical Questions of Students Training For Academic Olympics on Descriptive Geometry." Geometry & Graphics 5, no. 1 (April 17, 2017): 73–81. http://dx.doi.org/10.12737/25126.
Full textAbstract:
Academic Olympics on descriptive geometry are an important form of educational process in conditions when the discipline study’s teaching hours are reduced. The Academic Olympics allow identify talented students and motivate them to find possibilities on solutions for non-standard situations, to promote the formation of such qualities as persistence, endurance, accuracy. The Academic Olympics carrying requires not only organizational measures. Due to a low level of the geometrical knowledge gained at school, students need an additional theoretical training. Without it realizing of young people’s creative potential is impossible. The experience on carrying of the Academic Olympics on descriptive geometry at Bauman Moscow State Technical University and in other higher education institutions of this country allowed develop a technique of students training for such activities. In this paper scientific and methodical questions related to solution for tasks of increased complexity are considered. Simplification of solution algorithm and increase of its visualization can be reached by reduction of metric conditions to affine ones, and them, in turn — to projective ones. The attention in training for the Academic Olympics should be paid to the method of loci. School leavers know only about limited quantity of loci on a plane and have practically no ideas on the elementary loci in space. Therefore studying of loci on a plane and in space is necessary. For the method of loci the essence of simplification, decomposition and increments rules is revealed. If a return problem is solved more simply than a straight one, it is reasonable to use reverse ability method. Often the problem solution becomes simpler when using transformations, in particular, homothety and stretching (compression) transformations. The considered methods are illustrated by specific examples. Assimilation of the methods described in this publication isn't a complete guarantee of students’ success at the Academic Olympics, but, undoubtedly, it will be useful for them as will allow expand knowledge of the subject matter.
17
Dais, Dimitrios I. "Toric log del Pezzo surfaces with one singularity." Advances in Geometry 20, no. 1 (January 28, 2020): 121–38. http://dx.doi.org/10.1515/advgeom-2019-0009.
Full textAbstract:
AbstractThis paper focuses on the classification (up to isomorphism) of all toric log Del Pezzo surfaces with exactly one singularity, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective spaces.
18
Боровиков, И., Ivan Borovikov, Геннадий Иванов, Gennadiy Ivanov, Н. Суркова, and N. Surkova. "On Application of Transformations at Descriptive Geometry’s Problems Solution." Geometry & Graphics 6, no. 2 (August 21, 2018): 78–84. http://dx.doi.org/10.12737/article_5b55a35d683a33.30813949.
Full textAbstract:
This publication is devoted to the application of transformations at descriptive geometry’s problems solution. Using parametric calculus lets rationally select the number of transformations in the drawing. In Cartesian coordinates, on condition that an identical coordinate plane exists, the difference between parameters of linear forms, given and converted ones, is equal to the number of transformations in the composition. In affine space under these conditions, this difference is equal to two. Based on parameters calculation the conclusion is confirmed that the method of rotation around the level line, as providing the transformation of the plane of general position to the level plane, is a composition of two transformations: replacement of projections planes and rotation around the projection line. In various geometries (affine, projective, algebraic ones, and topology) the types of corresponding transformations are studied. As a result of these transformations are obtained affine, projective, bi-rational and topologically equivalent figures respectively. Such transformations are widely used in solving of applied problems, for example, in the design of technical surfaces of dependent sections. At the same time, along with transformation invariants, the simplicity of the algorithm for constructing of corresponding figures should be taken into account, with the result that so-called stratified transformations are preferred. A sign of transformation’s stratification is a value of dimension for a set of corresponding points’ carriers. This fact explains the relative simplicity of the algorithm for constructing the corresponding points in such transformations. In this paper the use of stratified transformations when finding the points of intersection of a curve with a surface, as well as in the construction of surfaces with variable cross-section shape are considered. The given examples show stratification idea possibilities for solving the problems of descriptive geometry.
19
Crooks, Peter, and Steven Rayan. "Some results on equivariant contact geometry for partial flag varieties." International Journal of Mathematics 27, no. 08 (July 2016): 1650066. http://dx.doi.org/10.1142/s0129167x1650066x.
Full textAbstract:
We study equivariant contact structures on complex projective varieties arising as partial flag varieties [Formula: see text], where [Formula: see text] is a connected, simply-connected complex simple group of type ADE and [Formula: see text] is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby’s classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of [Formula: see text]-equivariant vector bundles on [Formula: see text]. A byproduct of our argument is a canonical, global description of the unique [Formula: see text]-invariant contact structure on the isotropic Grassmannian of 2-planes in [Formula: see text].
20
Havlicek, Hans, Stefano Pasotti, and Silvia Pianta. "Automorphisms of a Clifford-like parallelism." Advances in Geometry 21, no. 1 (January 1, 2021): 63–73. http://dx.doi.org/10.1515/advgeom-2020-0027.
Full textAbstract:
Abstract We focus on the description of the automorphism group Γ∥ of a Clifford-like parallelism ∥ on a 3-dimensional projective double space (ℙ(HF ), ∥ ℓ , ∥ r ) over a quaternion skew field H (of any characteristic). We compare Γ∥ with the automorphism group Γ ℓ of the left parallelism ∥ ℓ , which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose ∥ in such a way that Γ∥ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ∥ ≠ ∥ ℓ .
21
Alexeev, Valery, Ron Donagi, Gavril Farkas, Elham Izadi, and Angela Ortega. "The uniformization of the moduli space of principally polarized abelian 6-folds." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 761 (April 1, 2020): 163–217. http://dx.doi.org/10.1515/crelle-2018-0005.
Full textAbstract:
AbstractStarting from a beautiful idea of Kanev, we construct a uniformization of the moduli space \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_{6}-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.
22
Волошинов, Д., and Denis Voloshinov. "Visual-Graphic Design of a Unitary Constructive Model to Solve Analogues For Apollonius Problem Taking into Account Imaginary Geometric Images." Geometry & Graphics 6, no. 2 (August 21, 2018): 23–46. http://dx.doi.org/10.12737/article_5b559c70becf44.21848537.
Full textAbstract:
The Apollonius problem on construction of circles, tangent to three arbitrary given circles of a plane, is one of classical geometry’s well-studied problems. The presented paper’s materials are directed at development a unified theory for Apollonius problem solving, taking into account it’s not only real, but also invisible complex-valued images. In the paper it has been demonstrated, that fundamental geometric structures, on which Apollonius problem is based on, are applicable not only to real, but also to complex-valued data, that makes possible to eliminate many exceptions, currently existing in it. In this paper Apollonius problem’s fundamental nature and its strong correlation with projective and quadratic geometric transformations has been disclosed. It has been proved that Apollonius problem and its analogues have a single solution method, in contrast to the prevailing idea that these problems can be solved only by separate particular methods. A concept of geometric experiment proposed by the author has allowed find out many previously unknown and discussed in this paper common factors, due to the set of many computational tests in the system Simplex for visual design of geometric models. In this paper is considered an example for solving an analogue of Apollonian problem for three-dimensional space, but proposed algorithm’s operation is universal, and it can be equally applied to solving similar problems in spaces of arbitrary dimensions. Obtained results demonstrate capabilities of methods for constructive modeling and multidimensional descriptive geometry in application to solving of complex mathematical problems, and determine the trends in development for automation systems of constructive geometric modeling.
23
Tudoreanu, Adrian, and Delia Drăgan. "Study on the Representation of Rotation Quadrics - The Ellipsoid." Advanced Engineering Forum 21 (March 2017): 426–33. http://dx.doi.org/10.4028/www.scientific.net/aef.21.426.
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The ellipsoid is part of the family of rotation quadrics, along with the paraboloid and hyperboloid. This paper presents a study on the representation of the ellipsoid surface using various representation manners from descriptive geometry: the orthogonal projection on two planes of projection (the Monge projection), the axonometric projection and the projection with elevations. Every type of representation is exemplified by drawings.
24
Guillou, Bertrand J. "The motivic fundamental group of the punctured projective line." Journal of K-Theory 7, no. 1 (April 7, 2010): 19–53. http://dx.doi.org/10.1017/is010003006jkt113.
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AbstractWe describe a construction of an object associated to the fundamental group of ℙ1 − {0, 1, ∞} in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products in the motivic cohomology of the ground field.
25
Ruzzi, Alessandro. "Geometrical description of smooth projective symmetric varieties with Picard number one." Transformation Groups 15, no. 1 (January 27, 2010): 201–26. http://dx.doi.org/10.1007/s00031-010-9074-9.
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Короткий, Виктор, Viktor Korotkiy, Е. Усманова, and E. Usmanova. "Second Order Curves on Computer Screen." Geometry & Graphics 6, no. 2 (August 21, 2018): 100–112. http://dx.doi.org/10.12737/article_5b55a829cee6c0.74112002.
Full textAbstract:
Modern computer graphics is based on methods of computational geometry. The curves and surfaces’ description is based on apparatus of spline functions, which became the main tool for geometric modeling. Methods of projective geometry are almost not applying. One of the reasons for this is impossibility to exactly construct a second-order curve passing through given points and tangent to given straight lines. To eliminate this defect a computer program for second order curves construction has been developed. The program performs the construction of second-order curve’s metric (center, vertices, asymptotes, foci) for following combinations: • The second-order curve is given by five points; • The second-order curve is given by five tangent lines; • The second-order curve is given by a point and two tangent lines with points of contact indicated on them; • The parabola is given by four tangent lines; • The parabola is given by four points. In this paper are presented algorithms for construction a metric for each combination. After construction the metric the computer program written in AutoLISP language and using geometrically exact projective algorithms which don’t require algebraic computations draws a second-order curve. For example, to construct vertices and foci of two parabolas passing through four given points, it is only necessary to draw an arbitrary circle and several straight lines. To construct a conic metric passing through five given points, it is necessary to perform only three geometrically exact operations: to construct an involution of conjugate diameters, to find the main axes and asymptotes; to note the vertices of desired second-order curve. Has been considered the architectural appearance of a new airport in Simferopol. It has been demonstrated that a terminal facade’s wavelike form can be obtained with a curve line consisting of conic sections’ areas with common tangent lines at junction points. The developed computer program allows draw second-order curves. The program application will promote the development of computer graphics’ tools and techniques.
27
Короткий and Viktor Korotkiy. "Plane Fields’ Quadratic Cremona Correspondence Set By Imaginary F-Points." Geometry & Graphics 5, no. 1 (April 17, 2017): 21–31. http://dx.doi.org/10.12737/25120.
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Birational (Cremona) correspondences of two planes П, П' or Cremona transformations on the combined plane П = П' represent an effective tool for design of smooth dynamic curves and two-dimensional lines. The simplest birational correspondence is a quadratic map Ω of plane fields on one another. In the projective definition of the quadratic correspondence can participate two pairs of imaginary complex conjugate F-points set as double points of elliptic involutions on the lines associated with the third pair of F-points. In this case, the imaginary projective F-bundles cannot be used for generation of points corresponding in Ω. A generic constructive algorithm for design of corresponding points in a quadratic mapping Ω(П ↔ П'), set both by real and imaginary F-points is proposed in this paper. The algorithm is based on the use of auxiliary projective correspondence Δ between the points of the planes П, П' and Hirst’s transformation Ψ with the center in the one of real F-points. A theorem on the existence of an invariant conic common to Ω and Δ mappings has been proved. Has been demonstrated a possibility for quadratic mapping’s presentation as a product of collinearity and Hirst’s transformation: Ω = ΔΨ. Has been considered a solution for auxiliary problems arising during the generic constructive algorithm’s implementation: buildup a conic section, that is incident to imaginary points, and plotting the corresponding points in collinearity set by imaginary points. It has been demonstrated that there are two or four possible versions of collinearity for plane fields П, П', set by with participation of the imaginary corresponding points, due to an uncertainty related to the order of their relative correspondence. Have been completely solved the problem of mapping a straight line in a conic section within the quadratic Cremona correspondance of fields П ≠ П', set by a pair of real fundamental points, and two pairs of imaginary ones. It has been demonstrated that in general case the problem has two solutions. The obtained results are useful for the development of the geometric theory related to imaginary elements and this theory’s application in linear and non-linear descriptive geometry, operating projective images of first and second orders.
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Wendt, Matthias. "Rationally trivial torsors in -homotopy theory." Journal of K-theory 7, no. 3 (May 16, 2011): 541–72. http://dx.doi.org/10.1017/is011004020jkt157.
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AbstractIn this paper, we show that rationally trivial torsors under split smooth linear algebraic groups induce fibre sequences in -homotopy theory. The results allow geometric proofs of stabilization results for unstable Karoubi-Villamayor K-theories and a description of the second -homotopy group of the projective line.
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Gorodezky, Marat. "The Dialectics of Creation and the Projective Structure of Space." Ideas and Ideals 13, no. 1-2 (March 19, 2021): 357–76. http://dx.doi.org/10.17212/2075-0862-2021-13.1.2-357-376.
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The article considers creationism as a historically relevant principle in the scientific and philosophic aspects denoting the ontological structure of the world. Outside of the religious interpretation, the author speaks of the dialectics of creation, which is revealed as an implicative connection of the one and nothing. Logical inversion (logical turn), acting from within this implicative connection, is postulated as the principle of a fundamental negation, which, according to the author, forms the true and dramatic essence of the world as a creation. The author distances himself from the widespread discussion between evolutionism and scientific creationism, stating that it does not correspond to the very subject of creationism, understood as the implication of a real from nothing. The author focuses on considering ‘nothing’ as a purely dialectical / metaphysical principle and relies partly on the Hegel’s dialectic of ‘being’ and ‘nothing’, and partly on the neoplatonic concept of the one. Rejecting the medieval interpretation of the temporal beginning and the Hegel’s identity, he deduces a scheme of the logical connection between the one and the difference, which postulates the inversion (turnover) forming the creation - the one and the difference disjunctively change places, the one becomes the real, and the difference out of the one becomes nothing. It is argued that this postulate, in particular, refutes the thesis about the ‘fall into sin’. In the second part of the article, a spatial-phenomenological hypothesis is presented: the author provides a description of the space as a geometrical-semantic plane (projective structure). This hypothesis follows from the phenomenological problem of the duality of a geometric object, which results in the problem of ontological transition between a point and a line (in the aporia of the Eleats) and the related problem of spatial congruence / parallelism. According to the author, the potential for solving these not essentially mathematical, but metaphysical questions is the projective geometry, in which parallel lines intersect at ‘point at infinity’, and space is complemented by the ‘plane at infinity’. The essence of the solution consists, firstly, in the assumption of the single plane, which underlies the transition, and secondly, in the description of the perceived world as a result of a specific turn over and closure of this plane, forming the projective structure. The key in this part is the demonstration of the surface of a three-dimensional object as a phenomenon of perceptual-semantic unfolding, which can be imagined as an action of consciousness, consistently reducing the usual scheme. An important aspect of considering the projective structure is the correlation with ‘the Plane’ by G. Deleuze. The general idea of the article is that the dialectical scheme of creation and the projective structure of the space coincide: the logical inversion (logical turn), acting in connection of the one and nothing, and projective structural turnover – are the same things.
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ANTONELLI, P., R. BRADBURY, V. KŘIVAN, and H. SHIMADA. "A DYNAMICAL THEORY OF HETEROCHRONY: TIME-SEQUENCING CHANGES IN ECOLOGY, DEVELOPMENT AND EVOLUTION." Journal of Biological Systems 01, no. 04 (December 1993): 451–87. http://dx.doi.org/10.1142/s0218339093000264.
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This work provides a foundation for a quantitative dynamical theory of heterochronic processes in the evolution of colonial invertebrate animals including Bryozoans, Siphonophores and Ants. These processes are environmentally induced changes in the time-sequencing of growth and development which can produce alterations in the morphotypes or castes within an individual colony. Motivation comes from Křivan’s theory of environmentally induced constraints on population densities for ecological interactions, but the present theory is second order with allometric production variables xi and population densities for morphotypes, Ni. We are able to unite ecological theory and the allometric form of the Wilson Ergonomic Theory via projective differential geometry and Wagner spaces which provide a natural description of environmentally induced time-sequencing changes altering the allometric curve of a species. Such changes define a model of heterochronic processes important in paleontology.
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ANIELLO, P., J. CLEMENTE-GALLARDO, G. MARMO, and G. F. VOLKERT. "CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES." International Journal of Geometric Methods in Modern Physics 07, no. 03 (May 2010): 485–503. http://dx.doi.org/10.1142/s0219887810004300.
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The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.
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Machale, Des, and H. S. M. Coxeter. "Projective Geometry." Mathematical Gazette 74, no. 467 (March 1990): 82. http://dx.doi.org/10.2307/3618883.
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Rota, Gian-Carlo. "Projective geometry." Advances in Mathematics 77, no. 2 (October 1989): 263. http://dx.doi.org/10.1016/0001-8708(89)90023-6.
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Mesnil, Romain, Cyril Douthe, Olivier Baverel, and Bruno Leger. "Marionette Meshes: Modelling free-form architecture with planar facets." International Journal of Space Structures 32, no. 3-4 (June 2017): 184–98. http://dx.doi.org/10.1177/0266351117738379.
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We introduce an intuitive method, called Marionette, for the modelling of free-form architecture with planar facets. The method takes inspiration from descriptive geometry and allows to design complex shapes with one projection and the control of elevation curves. The proposed framework achieves exact facet planarity in real time and considerably enriches previous geometrically constrained methods for free-form architecture. A discussion on the design of quadrilateral meshes with a fixed horizontal projection is first proposed, and the method is then extended to various projections and patterns. The method used is a discrete solution of a continuous problem. This relation between smooth and continuous problem is discussed and shows how to combine the marionette method with modelling tools for smooth surfaces, like non-uniform rational basis spline or T-splines. The result is a versatile tool for shape modelling, suited to engineering problems related to free-form architecture.
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Левкин, Ю., and Yu Levkin. "Five-Dimensional Two-Oktantal Epure Nomogram." Geometry & Graphics 5, no. 4 (December 13, 2017): 44–51. http://dx.doi.org/10.12737/article_5a17fecf2feac9.18123975.
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Multidimensional experimental tasks with interdependent physical quantities can't be characterized by use of flat two-dimensional plots. Nomograms of new type solve such tasks. In this paper have been presented the nomograms with systematized both axes and planes. At construction of such models it is required a clear separation of all parametrial variable on arguing and functional ones. Nearby axes of interdependent parameters should lie alongside. Each axonometric cell should have a resultant indicator in the form of full size’s geometrical image. For the optimum choice of graphic execution on tabular data with four or five parameters, in the present paper is offered a method of its realization by means of two-oktantal nomogram. Justifications for this method have been presented in the paper. The method itself is based on descriptive geometry’s opportunities expansion at the solution of technical tasks by means of multidimensional geometry. The main lever for the task implementation is, certainly, communication lines. Formerly known from descriptive geometry such concepts as plane of reference, horizontally projecting surface, on the one hand, and pointed measurement of all experimental parameters on the other hand, provides to the nomogram possibility of its understanding for genesis in physical processes. Based on similarity of adjacent oktantal cells having the general axis are plotting two oktantal axonometric nomograms, creating interdependence between parameters by means of communication lines. This method opens a possibility for understanding of physical processes transformation. In this paper have been presented two graphic models of two oktantal nomograms competing for the right to be used by force of theirs optimal advantages. Absolute values of parameters are the real ones, taken from papers in "News of Higher Educational Institutions. North Caucasus Region" journal. Technical Sciences. No. 3, pp. 77–83, and No. 2, pp. 112–119. 2016.
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AGORE, A. L., S. CAENEPEEL, and G. MILITARU. "THE CENTER OF THE CATEGORY OF BIMODULES AND DESCENT DATA FOR NONCOMMUTATIVE RINGS." Journal of Algebra and Its Applications 11, no. 06 (November 14, 2012): 1250102. http://dx.doi.org/10.1142/s0219498812501022.
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Let A be an algebra over a commutative ring k. We compute the center of the category of A-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical A-coring, Yetter–Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. We provide several applications: for instance, if A is finitely generated projective over k then the category of left End k(A)-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of A. As another application, new families of solutions for the quantum Yang–Baxter equation are constructed: they are canonical maps Ω associated to any right comodule over the Sweedler canonical coring A ⊗ A and satisfy the condition Ω3 = Ω. Explicit examples are provided.
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Chaput, Pierre-Emmanuel. "Geometry over composition algebras: Projective geometry." Journal of Algebra 298, no. 2 (April 2006): 340–62. http://dx.doi.org/10.1016/j.jalgebra.2006.02.008.
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Kanatani, Kenichi. "Computational projective geometry." CVGIP: Image Understanding 54, no. 3 (November 1991): 333–48. http://dx.doi.org/10.1016/1049-9660(91)90034-m.
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Calderbank, David, Michael Eastwood, Vladimir Matveev, and Katharina Neusser. "C-projective geometry." Memoirs of the American Mathematical Society 267, no. 1299 (September 2020): 0. http://dx.doi.org/10.1090/memo/1299.
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CHERVOV, A., and D. TALALAEV. "HITCHIN SYSTEMS ON SINGULAR CURVES II: GLUING SUBSCHEMES." International Journal of Geometric Methods in Modern Physics 04, no. 05 (August 2007): 751–87. http://dx.doi.org/10.1142/s0219887807002284.
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In this paper we continue our studies of Hitchin systems on singular curves (started in [1]). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec {f ∈ ℂ[z] : f(A(∊)) = f(B(∊)); ∊N = 0}, where A(∊), B(∊) are arbitrary polynomials). The most simple examples are the generalized cusp curves which are projectivizations of Spec {f ∈ ℂ[z] : f′(0) = f″(0) = ⋯ fN-1(0) = 0}. We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials A(∊), B(∊) defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop r-matrix formalism for the functions on the truncated loop group GLn(ℂ[z]), zN = 0. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space.
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Сальков and Nikolay Sal'kov. "Dupin Cyclide and Second-Order Curves. Part 1." Geometry & Graphics 4, no. 2 (June 18, 2016): 19–28. http://dx.doi.org/10.12737/19829.
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Making smooth shapes of various products is caused
by the following requirements: aerodynamic, structural, aesthetic,
etc. That’s why the review of the topic of second-order curves is
included in many textbooks on descriptive geometry and engineering
graphics. These curves can be used as a transition from the one
line to another as the first and second order smoothness. Unfortunately,
in modern textbooks on engineering graphics the building of Konik
is not given. Despite the fact that all the second-order curves are banded by a single analytical equation, geometrically they unites
by the affiliation of the quadric, projective unites by the commonality
of their construction, in the academic literature for each of
these curves is offered its own individual plot. Considering the
patterns associated with Dupin cyclide, you can pay attention to
the following peculiarity: the center of the sphere that is in contact
circumferentially with Dupin cyclide, by changing the radius of the
sphere moves along the second-order curve. The circle of contact
of the sphere with Dupin cyclide is always located in a plane passing
through one of the two axes, and each of these planes intersects
cyclide by two circles. This property formed the basis of the graphical
constructions that are common to all second-order curves. In
addition, considered building has a connection with such transformation
as the dilation or the central similarity. This article considers
the methods of constructing of second-order curves, which are
the lines of centers tangent of the spheres, applies a systematic
approach.
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Dillon, Meighan. "Projective Geometry for All." College Mathematics Journal 45, no. 3 (May 2014): 169–78. http://dx.doi.org/10.4169/college.math.j.45.3.169.
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López Peña, Javier, and Oliver Lorscheid. "Projective geometry for blueprints." Comptes Rendus Mathematique 350, no. 9-10 (May 2012): 455–58. http://dx.doi.org/10.1016/j.crma.2012.05.001.
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Gupta, K. C., and Suryansu Ray. "Fuzzy plane projective geometry." Fuzzy Sets and Systems 54, no. 2 (March 1993): 191–206. http://dx.doi.org/10.1016/0165-0114(93)90276-n.
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Savel'ev, Yu, and Elena Cherkasova. "Computational Graphics in Solving of Non-Traditional Engineering Problems." Geometry & Graphics 8, no. 1 (April 20, 2020): 33–44. http://dx.doi.org/10.12737/2308-4898-2020-33-44.
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Based on the published data, the essence of computational graphics has been laid down. Have been reported examples of new results obtained only through accurate computer constructions and measurements. The work content is a presentation of new ways to achieve the goal by solving non-traditional engineering problems. The author's method of projection with time-stamps, which, in fact, is a computer descriptive geometry, allows solve multi-parameter (not to be confused with multi-dimensional) problems with 9 variables [1–3; 13]. The author’s method of two-axis equal-sized evolvement [11; 12; 17] allows quantitatively measurements of solid angles. The addition of trigonometric functions (sinuses, sinusoids, etc.) can also be considered as a novelty [10; 11; 14]. At the junction of analytic (AG) and descriptive geometries have been calculated parameters of dodecahedron and has been given its mathematical description. In the traditional AG task, the required parameters have been calculated graphically, including a point’s speed of movement. Has been presented the author’s method for determining the instantaneous center in theoretical mechanics. For the first time, the equality of the angles of rotation for points and the link as a whole has been established, and a continuous centroid has been built. By decomposition of vectors a new way for summing up theirs vertical projections has been demonstrated. The developed method of projections with time-stamps allows simultaneously consider such parameters as spatial coordinates of moving objects (two or more) in time, their speeds and even sizes, including the variable ones. Has been shown the possibility for graphical programming while solving systems of equations, as well as for graphical solution of algebraic and stereometric problems. This publication aims to disseminate computer methods for engineering problems solving.
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Burn, Bob, Lars Kadison, and Matthias T. Kromann. "Projective Geometry and Modern Algebra." Mathematical Gazette 80, no. 488 (July 1996): 446. http://dx.doi.org/10.2307/3619609.
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Grigorenko, A. N. "Geometry of projective Hilbert space." Physical Review A 46, no. 11 (December 1, 1992): 7292–94. http://dx.doi.org/10.1103/physreva.46.7292.
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Safari, R., N. Narasimhamurthi, M. Shridhar, and M. Ahmadi. "Document registration using projective geometry." IEEE Transactions on Image Processing 6, no. 9 (September 1997): 1337–41. http://dx.doi.org/10.1109/83.623198.
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Gros, P., R. Hartley, R. Mohr, and L. Quan. "How Useful is Projective Geometry?" Computer Vision and Image Understanding 65, no. 3 (March 1997): 442–46. http://dx.doi.org/10.1006/cviu.1996.0496.
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Delphenich, D. H. "Projective geometry and special relativity." Annalen der Physik 15, no. 3 (March 15, 2006): 216–46. http://dx.doi.org/10.1002/andp.200510179.
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