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Journal articles on the topic 'Theorganization of space on the plane'

Author: Grafiati

Published: 2 June 2025

Last updated: 22 June 2025

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1

Висікайло, Т. В. "Використання роботи над пейзажем у процесі формування просторового мислення студентів середніх художньо-освітніх закладів". Педагогіка та психологія : збірник наукових праць , № 49 (25 червня 2015): 165–73. https://doi.org/10.5281/zenodo.18994.

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Landscape has the considerable value on the training of art schools students. The uniqueness of the landscape art is the ability to solve complex graphical, colorful, composite tasks in a simple, understandable form. In the process of training the interest to creative comprehension of provided educational material is stimulating, a stable internal motivation of students to mastering new knowledge is forming. The expediency of using the landscape to form spatial thinking of students of art schools is the main task of training the secondary art institutions specialists. The effectiveness of the raised problems solution tied with the usage in the educational process model observations, the performance of graphics and color model sketches, combined with the study of the predecessor’s experience. Landscape working helps students clearly demonstrates the possibilities of organization of spatial relations at the plane, variants of harmonious placement of objects in the image space, ascertainment relationship between the method of organization of spatial relations on the plane and its impact on the viewer. In the formation of spatial thinking, as the ability to creative interpretation of reality, knowledge and skills, gained while working on the landscape, should help the students. 

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2

OHKAMI, Yoshiaki. "Space Transportation System and Space Plane." Journal of the Society of Mechanical Engineers 94, no. 871 (1991): 478–82. http://dx.doi.org/10.1299/jsmemag.94.871_478.

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3

Wintle, H. J. "Point-Plane and Edge-Plane Space Charge Limited Flows." IEEE Transactions on Electrical Insulation EI-21, no. 3 (1986): 365–73. http://dx.doi.org/10.1109/tei.1986.349078.

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4

Neffe, Jürgen. "European space programme: Germany plans a space plane." Nature 323, no. 6085 (1986): 195. http://dx.doi.org/10.1038/323195b0.

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5

Jackson, Thomas A. "Power for a Space Plane." Scientific American 295, no. 2 (2006): 56–63. http://dx.doi.org/10.1038/scientificamerican0806-56.

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6

Schmidt, L. C. "Single-chorded Plane Space Trusses." International Journal of Space Structures 1, no. 2 (1985): 69–74. http://dx.doi.org/10.1177/026635118500100201.

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A discussion is presented of the mechanics of the plate-like behaviour of certain space trusses that possess a single layer of chords. The web systems are formed out of the chord plane, and consideration is given to web systems that are placed on one or both sides of the chord plane. It is recognized that a torsional mode of behaviour is utilized for the double-layer web system for carrying transverse load. A combination of flexural and torsional modes is utilized for the single-layer system.

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7

Dickman, Steven. "Space plane spawns hypersonic plans." Nature 339, no. 6224 (1989): 409. http://dx.doi.org/10.1038/339409c0.

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8

Ottmann, Thomas. "Space-economical plane-sweep algorithms." Computer Vision, Graphics, and Image Processing 32, no. 1 (1985): 143. http://dx.doi.org/10.1016/0734-189x(85)90013-1.

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9

Ottmann, Thomas, and Derick Wood. "Space-economical plane-sweep algorithms." Computer Vision, Graphics, and Image Processing 34, no. 1 (1986): 35–51. http://dx.doi.org/10.1016/0734-189x(86)90046-0.

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10

Gaikwad, Vidya, and T. M. Karade. "Plane symmetric higher dimensional space-times." Acta Physica Hungarica 67, no. 3-4 (1990): 259–62. http://dx.doi.org/10.1007/bf03155806.

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11

Dickman, Steven. "Full speed ahead with space plane." Nature 338, no. 6211 (1989): 105. http://dx.doi.org/10.1038/338105c0.

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12

Nouri-Zonoz, M., and A. R. Tavanfar. "Plane-symmetric analogue of NUT space." Classical and Quantum Gravity 18, no. 20 (2001): 4293–302. http://dx.doi.org/10.1088/0264-9381/18/20/308.

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13

Cook, William. "Cutting-plane proofs in polynomial space." Mathematical Programming 47, no. 1-3 (1990): 11–18. http://dx.doi.org/10.1007/bf01580849.

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14

NISHIO, Masatomi, Kenji KAI, and Shinji SEZAKI. "711 Flow around Hypersonic Space Plane." Proceedings of Conference of Chugoku-Shikoku Branch 2001.39 (2001): 263–64. http://dx.doi.org/10.1299/jsmecs.2001.39.263.

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15

Abío, Ignasi, Maria Alberich-Carramiñana, and Víctor González-Alonso. "The Ultrametric Space of Plane Branches." Communications in Algebra 39, no. 11 (2011): 4206–20. http://dx.doi.org/10.1080/00927872.2010.521934.

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16

MAITA, Masataka. "Special issue : space plane plan in Japan and its elemental technology.System concepts of the space plane." Journal of the Japan Society for Aeronautical and Space Sciences 39, no. 454 (1991): 571–79. http://dx.doi.org/10.2322/jjsass1969.39.571.

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17

Tanhayi, M. Reza. "Particle creation in global de Sitter space: Bulk space consideration." International Journal of Modern Physics D 24, no. 07 (2015): 1550052. http://dx.doi.org/10.1142/s0218271815500522.

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Recently in [P. R. Anderson and E. Mottola, Phys. Rev. D 89 (2014) 104039, arXiv:1310.1963 [gr-qc] and P. R. Anderson and E. Mottola, Phys. Rev. D 89 (2014) 104038, arXiv:1310.0030 [gr-qc].], it was shown that global de Sitter space is unstable even to the massive particle creation with no self-interactions. In this paper, we study the instability by making use of the coordinate-independent plane wave in de Sitter space. Within this formalism, we show that the previous results of instability of de Sitter space due to the particle creation can be generalized to higher-spin fields in a straightforward way. The so-called plane wave is defined globally in de Sitter space and de Sitter invariance is manifest since such modes are deduced from the group theoretical point of view by means of the Casimir operators. In fact, we employ the symmetry of embedding space namely the 4 + 1-dimensional flat space to write the field equations and the solutions can be obtained in terms of the plane wave in embedding space.

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18

SHARIF, M., and ZAHID AHMAD. "PLANE SYMMETRIC GRAVITATIONAL COLLAPSE." International Journal of Modern Physics A 23, no. 01 (2008): 181–88. http://dx.doi.org/10.1142/s0217751x0803797x.

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In this paper, we derive the general formulation by considering two arbitrary plane symmetric space–times using Israel's method. As an example, we apply this formulation to known plane symmetric space–times. We take the Taub's static metric in the interior region whereas Kasner's nonstatic metric in the exterior region. It is shown that the plane collapses in some cases whereas it expands in some others.

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19

Kahriman, Elif Altintas. "Fuzzy collineations of 3-dimensional fuzzy projective space from 4-dimensional fuzzy vector space." AIMS Mathematics 9, no. 9 (2024): 26182–94. http://dx.doi.org/10.3934/math.20241279.

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<p>In this paper, the fuzzy counterparts of the collineations defined in classical projective spaces are defined in a 3-dimensional fuzzy projective space derived from a 4-dimensional fuzzy vector space. The properties of fuzzy projective space $ (\lambda, \mathcal{S}) $ left invariant under the fuzzy collineations are characterized depending on the membership degrees of the given fuzzy projective space and also depending on the pointwise invariant of the lines. Moreover, some relations between membership degrees of the fuzzy projective space are presented according to which are of the base point, base line, and base plane invariant under a fuzzy collineation. Specifically, when all membership degrees of $ (\lambda, \mathcal{S}) $ are distinct, the base point, base line, and base plane of $ (\lambda, \mathcal{S}) $ are invariant under the fuzzy collineation $ \bar{f} $. Conversely, if none of the base point, base line, or base plane remain invariant, then the system becomes crisp in $ (\lambda, \mathcal{S}) $. Additionally, some relations between the membership degrees of the fuzzy projective space, concerning the invariance of the base point, base line, and base plane, are presented.</p>

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20

Kotlyar, V. V., A. A. Kovalev, and A. G. Nalimov. "Optical phase singularities and superluminal motion in unbounded space." Computer Optics 5, no. 45 (2021): 654–60. http://dx.doi.org/10.18287/2412-6179-co-879.

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In this paper, we summarize a remarkable result obtained by Soskin et al. in Phys Rev A 56, 4064 (1997). We show that for an on-axis superposition of two different-waist Laguerre-Gauss beams with numbers (0, n) and (0, m), the topological charge equals TC=m up to a plane where the waist radii become the same, given that the beam (0, m) has a greater waist radius, changing to TC=n after this plane. This occurs because in the initial plane the superposition has an on-axis op-tical vortex with TC=m and on different axis-centered circles there are (n – m) vortices with TC= +1 and (n – m) vortices with TC= –1. On approaching the above-specified plane, the vortices with TC= -1 "depart" to infinity with a higher-than-light speed, with the TC of the total beam becoming equal to TC=n. If, on the contrary, the beam (0, m) has a smaller waist, then the total TC equals n on a path from the initial plane up to a plane where the waist radii become the same, changing to TC=m after the said plane. This occurs because after the said plane, n–m vortices with TC= –1 "arrive" from infinity with a higher-than-light speed.

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21

Kotlyar, V. V., A. A. Kovalev, and A. G. Nalimov. "Optical phase singularities and superluminal motion in unbounded space." Computer Optics 5, no. 45 (2021): 654–60. http://dx.doi.org/10.18287/2412-6179-co-879.

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In this paper, we summarize a remarkable result obtained by Soskin et al. in Phys Rev A 56, 4064 (1997). We show that for an on-axis superposition of two different-waist Laguerre-Gauss beams with numbers (0, n) and (0, m), the topological charge equals TC=m up to a plane where the waist radii become the same, given that the beam (0, m) has a greater waist radius, changing to TC=n after this plane. This occurs because in the initial plane the superposition has an on-axis op-tical vortex with TC=m and on different axis-centered circles there are (n – m) vortices with TC= +1 and (n – m) vortices with TC= –1. On approaching the above-specified plane, the vortices with TC= -1 "depart" to infinity with a higher-than-light speed, with the TC of the total beam becoming equal to TC=n. If, on the contrary, the beam (0, m) has a smaller waist, then the total TC equals n on a path from the initial plane up to a plane where the waist radii become the same, changing to TC=m after the said plane. This occurs because after the said plane, n–m vortices with TC= –1 "arrive" from infinity with a higher-than-light speed.

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22

Krasic, Sonja, and Biserka Markovic. "Graphic representation of a triaxial ellipsoid by means of a sphere in general collinear spaces." Facta universitatis - series: Architecture and Civil Engineering 9, no. 2 (2011): 269–75. http://dx.doi.org/10.2298/fuace1102269k.

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For graphic representation of the projective creations, such as the quadrics (II degree surfaces) in projective, general collinear spaces, it is necessary to firstly determine the characteristic parameters, such as: vanishing planes, axes and centers of space. An absolute conic of a space is an imaginary conic, residing in the infinitely distant plane of that space. The common elements of the absolute conic and infinitely distant conic of a quadric in the infinitely distant plane of that space are the autopolar triangle and two double straight lines which are always real and it is necessary to use the common elements of their associated pair of conics in the vanishing plane of the associated space. The quadric axes are passing through the apices of the autopolar triangle, and they are important for graphic representation of the quadrics. In order to map a sphere in the first space into the triaxial ellipsoid in the second space, it is necessary to select a sphere so that its center is not on the axis of that space and that it intersects the vanishing plane of the second space along the imaginary circumference, which is in general position with the figure of the absolute conic of the second space (the associated pair of conics in the vanishing plane).

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23

Babourova, O. V., B. N. Frolov, M. S. Khetzeva, and D. V. Kushnir. "The structure of the curvature tensor of plane gravitational waves." Journal of Physics: Conference Series 2081, no. 1 (2021): 012014. http://dx.doi.org/10.1088/1742-6596/2081/1/012014.

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Abstract Plane gravitational waves in the Riemann space of General Relativity is considered. The criterion of plane gravitational waves is used based on the analogy between plane gravitational and electromagnetic waves. The Theorem is proved that the action of the Lie derivative on the plane wave curvature 2-form in the direction of the vector generating the invariance group of this wave in the Riemann space is equal to zero. It is justified that the gravitational waves can be used to transmit information in the Riemann space.

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24

Luo, Shihao, Naigang Cui, Xiaowei Wang, Youhua Fan, and Haitao Gu. "The Stability Analysis of a Tether for a Segmented Space Elevator." Aerospace 9, no. 7 (2022): 376. http://dx.doi.org/10.3390/aerospace9070376.

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The space elevator system is a space tether system used to solve low-cost space transportation. Its high efficiency, large load and other characteristics have broad application prospects in the aerospace field. The stability analysis is the foundation of the space elevator system research. Based on the new segment space elevator system model, in this paper, the stability of the system at the equilibrium point is analyzed by Lyapunov stability theory; And based on the criterion that the change rate of the system restoring torque and the anchor point tension are greater than 0, the maximum offset angle of the system inside and outside the equatorial plane is analyzed. The results show that the segment space elevator is stable near the equilibrium point; The maximum deflection angle of the space elevator inside and outside the equatorial plane is related to the design stress of the anchor point; When the space elevator is offset outside the equatorial plane, it will only lose stability because the restoring torque reaches the maximum value; When the space elevator is offset in the equatorial plane, and due to the design stress of the anchor point is small, it will lose stability because the tensile force of the anchor point is reduced to 0, and when the design stress of the anchor point is large, it will lose stability because the recovery torque reaches the maximum value; The stability of the space elevator outside the equatorial plane is better than that in the equatorial plane.

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25

Matsievsky, S., and S. Aleshnikov. "CONCHOIDAL TRANSFORM OF SPACE." Znanstvena misel journal, no. 97 (December 30, 2024): 38–47. https://doi.org/10.5281/zenodo.14575439.

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The conchoidal transformation of Euclidean space are defined. Under certain constraints, they form an Abelian group, and their composition acts on a single straight line or plane. The vertices of limaçons with a common base lie on the straight line and the cissoid of Diocles, and the inflection points on a generalized piriform quartic. The composition of conchoidal transformations of the plane forms seven families of congruent triangles, or two limaçons with one base circle form two families of congruent parallelograms. Under certain constraints, the composition of conchoid transformations of space is a conchoid transformation, and conchoid transformations can form a non-Abelian group.

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26

Kovačević, Domagoj, Stjepan Meljanac, Andjelo Samsarov та Zoran Škoda. "Hermitian realizations of κ-Minkowski space–time". International Journal of Modern Physics A 30, № 03 (2015): 1550019. http://dx.doi.org/10.1142/s0217751x15500190.

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General realizations, star products and plane waves for κ-Minkowski space–time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.

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27

Johnsen, Trygve. "Plane projections of a smooth space curve." Banach Center Publications 36, no. 1 (1996): 89–110. http://dx.doi.org/10.4064/-36-1-89-110.

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28

Astala, Kari, and Juha Heinonen. "On quasiconformal rigidity in space and plane." Annales Academiae Scientiarum Fennicae. Series A. I. Mathematica 13 (1988): 81–92. http://dx.doi.org/10.5186/aasfm.1988.1301.

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29

Hafner, Christian, and Bernd Bickel. "The design space of plane elastic curves." ACM Transactions on Graphics 40, no. 4 (2021): 1–20. http://dx.doi.org/10.1145/3476576.3476697.

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30

Hafner, Christian, and Bernd Bickel. "The design space of plane elastic curves." ACM Transactions on Graphics 40, no. 4 (2021): 1–20. http://dx.doi.org/10.1145/3450626.3459800.

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31

Turchyn, Ihor, and Olga Turchyn. "TRANSIENT PLANE WAVES IN MULTILAYERED HALF-SPACE." Acta Mechanica et Automatica 7, no. 1 (2013): 53–57. http://dx.doi.org/10.2478/ama-2013-0010.

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Abstract Considered the dynamic problem of the theory of elasticity for multilayered half-space. Boundary surface of inhomogeneous half-space loaded with normal load, and the boundaries of separation layers are in conditions of ideal mechanical contact. The formulation involves non-classical separation of equations of motion using two functions with a particular mechanical meaning volumetric expansion and function of acceleration of the shift. In terms of these functions obtained two wave equation, written boundary conditions and the conditions of ideal mechanical contact of layers. Using the Laguerre and Fourier integral transformations was obtained the solution of the formulated problem. The results of the calculation of the stress-strain state in the half-space with a coating for a local impact loading are presented.

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32

Morino, Yoshiki. "Materials and Joining in Space Plane Technologies." Journal of the Japan Welding Society 62, no. 8 (1993): 617–22. http://dx.doi.org/10.2207/qjjws1943.62.617.

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33

INOUE, Yasutoshi, and Yukimitsu YAMAMOTO. "Hypersonic aerodynamic heating studies for space plane." Journal of the Japan Society for Aeronautical and Space Sciences 38, no. 435 (1990): 204–15. http://dx.doi.org/10.2322/jjsass1969.38.204.

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34

Robinson, James J. "Getting the Space Plane off the Ground." JOM 39, no. 7 (1987): 8–9. http://dx.doi.org/10.1007/bf03258031.

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35

Xu, Lei, Jody C. Leng, Edward R. Mariano, and Ban C. H. Tsui. "Erector spinae plane: a collapsible potential space." Regional Anesthesia & Pain Medicine 45, no. 7 (2019): 562–63. http://dx.doi.org/10.1136/rapm-2019-101107.

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36

Li, Jianzeng. "Space‐times with plane‐symmetric scalar waves." Journal of Mathematical Physics 33, no. 10 (1992): 3506–8. http://dx.doi.org/10.1063/1.529901.

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37

Frenkel, Igor B., and Hyun Kyu Kim. "Quantum Teichmüller space from the quantum plane." Duke Mathematical Journal 161, no. 2 (2012): 305–66. http://dx.doi.org/10.1215/00127094-1507390.

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38

Dirl, R., K. Payer, and B. L. Davies. "Symmetrized plane waves: 1. symmorphic space groups." Computer Physics Communications 98, no. 1-2 (1996): 52–72. http://dx.doi.org/10.1016/0010-4655(96)00072-0.

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39

Dirl, R., K. Payer, and B. L. Davies. "Symmetrized plane waves: 3. nonsymmorphic space groups." Computer Physics Communications 98, no. 1-2 (1996): 83–97. http://dx.doi.org/10.1016/0010-4655(96)00074-4.

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40

Nandi, Santanu. "On dynamics of $\lambda + \tan z^2 $." Ukrains’kyi Matematychnyi Zhurnal 77, no. 3 (2025): 235–36. https://doi.org/10.3842/umzh.v77i3.8365.

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UDC 517.9 We propose a new family of transcendental meromorphic functions $\lambda + \tan z^2$ for $ \lambda \in \mathbb C$ and study the dynamics of the family of functions. We explore both the dynamical plane ($z$-plane) and the parameter plane ($\lambda$-plane). We show that, in the dynamical plane, there are no Herman rings, and the Julia set forms a Cantor set when the parameter lies within the unbounded hyperbolic components. In addition, it is proved that these unbounded hyperbolic components are the only available components distributed over the four quadrants of the parameter space in the complex plane. Conversely, it is shown that the Julia set is connected for the maps whose parameter lies within the remaining hyperbolic components of the parameter space. We also perform the comprehensive analysis of the combinatorial structure of both the parameter space and the dynamical plane for this family of transcendental meromorphic maps.

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41

Zhang, Yu Huan. "Print Ads "Three-Dimensional" Form of Research." Advanced Materials Research 912-914 (April 2014): 1892–95. http://dx.doi.org/10.4028/www.scientific.net/amr.912-914.1892.

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In the two-dimensional plane to refine core design of visual elements and will direct the important influence on the processing of space, space of display design also inevitable need considering the visual elements and plane of coordination and unity. The multimedia advertising originality and product value very clever combination of strong visual impact. To regard the "visibility" disorders, caused by the wrong is increasingly being used, to increase the rate of visual attention and increasing the effect of the ornamental interest. Plane outdoor advertising performance breakthrough in the plane of the two-dimensional space, create a new and unique visual effect.

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42

Glubokov, Andrey. "Ideals on the Quantum Plane’s Jet Space." Mathematics 8, no. 3 (2020): 352. http://dx.doi.org/10.3390/math8030352.

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The goal of this paper is to introduce some rings that play the role of the jet spaces of the quantum plane and unlike the quantum plane itself possess interesting nontrivial prime ideals. We will prove some results (Theorems 1–4) about the prime spectrum of these rings.

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43

Zhao, Jun, Feicheng Wang, Bai Yang, and Bin Ma. "Seismic Behaviour of CFST Space Intersecting Nodes in an Oblique Mesh." Applied Sciences 13, no. 10 (2023): 5943. http://dx.doi.org/10.3390/app13105943.

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The design of intersecting nodes in high-rise oblique mesh structures is a critical issue. The existing research on the intersecting nodes of oblique meshes mainly focuses on plane intersecting nodes and monotonic axial compression loads. The plane intersecting nodes cannot consider the contribution of the node’s out-of-plane angle and floor beam to the node’s out-of-plane stiffness in actual structures. In this paper, numerical analysis using ABAQUS was conducted to investigate the mechanical performance of space intersecting nodes of oblique meshes (OMSIN) under cyclic axial tension and compression loads, to provide a reference for the engineering application of oblique mesh structures in seismic regions. Six parameters were considered: the space intersecting angle, the plane angle symmetry coefficient, the plane intersecting angle, the out-of-plane constraint restraint, the steel content of the cross-section, and the concrete strength. The study showed that changes in the thickness of the steel tube wall are unfavourable for the uniform transmission of stress. Increasing the space intersecting angle significantly weakened the seismic performance, and the space angle affects the failure mode of the node. Asymmetric arrangements of the upper and lower plane angles caused nonlinear development of out-of-plane. The ultimate load and overall compressive stiffness of the specimen were positively correlated with the plane angle, and vertical constraints should be applied to the node position of components with plane angles greater than or equal to 70°. The out-of-plane constraint was a key factor affecting the seismic performance of the node, and it was proportional to the ultimate load of the component. In structural design, if the aim is to improve the mechanical performance of the component by increasing the steel content, more enormous out-of-plane constraints should be set to control plane external displacement strictly. The concrete strength is proportional to the ultimate axial load and axial stiffness, and its influence on the mechanical performance in the axial tension direction is not significant. Finally, a dimensionless skeleton curve model of the node was established. The existing formula for the bearing capacity of CFST columns was fitted to obtain the calculation formula for the axial yield and ultimate load of the OMSIN under cyclic loads.

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44

Krori, K. D., Ranjana Choudhury, and J. C. Sarmah. "Stability of trajectories in Ernst space–time." Canadian Journal of Physics 64, no. 11 (1986): 1455–57. http://dx.doi.org/10.1139/p86-258.

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In this paper we show that stable trajectories of charged particles, neutral particles, photons, and tachyons occur in the equatorial plane of the Ernst space–time. We also present typical bound trajectories of these particles in the equatorial plane (θ = π/2) as well as on an r = constant surface.

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45

Dubanov, A. A. "Kinematic model of the parallel convergence method in space." Journal of Physics: Conference Series 2182, no. 1 (2022): 012004. http://dx.doi.org/10.1088/1742-6596/2182/1/012004.

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Abstract In this article, the implementation of the method of parallel convergence in space in a computer mathematics system is considered and discussed. In this method, the pursuer’s velocity vector is directed arbitrarily. The pursuer’s trajectory gradually approaches movement in the plane formed by the line connecting the initial positions of the pursuer and the target, and the velocity vector. In this task, the target moves uniformly and rectilinearly. The pursuer moves evenly. The points of the pursuer’s trajectory are calculated sequentially. They are being the result of the intersection of the plane containing the line of sight, sphere and cone. As we approach the plane where the target is moving, the algorithm for calculating the trajectory points changes. Now the point of the pursuer’s trajectory is the result of the intersection of the sphere, the plane of movement of the target and the plane containing the line of sight.

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46

Savas, Murat, Baki Karliga, and Atakan T. Yakut. "Orthogonal Projections Based on Hyperbolic and Sphericaln-Simplex." Advances in Mathematical Physics 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/808250.

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Orthogonal projection along a geodesic to the chosenk-plane is introduced using edge and Gram matrix of ann-simplex in hyperbolic or sphericaln-space. The distance from a point tok-plane is obtained by the orthogonal projection. It is also given the perpendicular foot from a point tok-plane of hyperbolic and sphericaln-space.

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47

Alghanemi, Azeb, and Peter Giblin. "On geometry of the midlocus associated to a smooth curve in plane and space." Filomat 32, no. 8 (2018): 2977–90. http://dx.doi.org/10.2298/fil1808977a.

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The singularities of the midpoint map associated to a smooth plane curve, which is a map from the plane to the plane, are classified. The midlocus associated to a regular space curve is introduced. The geometric conditions for the midlocus of a space curve to have a crosscap or an S?1 singularities are investigated. A more general map, the ?-point map, associated to a space curve is introduced and many known surface singularities are realized as a special cases of this construction.

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48

PYO, JUNCHEOL, and KEOMKYO SEO. "SPACELIKE CAPILLARY SURFACES IN THE LORENTZ–MINKOWSKI SPACE." Bulletin of the Australian Mathematical Society 84, no. 3 (2011): 362–71. http://dx.doi.org/10.1017/s0004972711002528.

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AbstractFor a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz–Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilical point, which was developed by Choe [‘Sufficient conditions for constant mean curvature surfaces to be round’, Math. Ann.323(1) (2002), 143–156]. Using the concept of the rotation index at the interior and boundary umbilical points and applying the Poincaré–Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than four vertices in a domain of $\Bbb L^3$ bounded by (spacelike or timelike) totally umbilical surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover, we prove that the only immersed spacelike disk type capillary surface inside a de Sitter surface in $\Bbb L^3$ is part of (spacelike) plane or a hyperbolic plane.

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Belova, Е., and O. Belova. "About an analogue of Neifeld’s connection on the space of centred planes with one-index basic-fibre forms." Differential Geometry of Manifolds of Figures, no. 50 (2019): 41–47. http://dx.doi.org/10.5922/0321-4796-2019-50-6.

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This research is realized by Cartan — Laptev method (with prolongations and scopes, moving frame and exterior forms). In this paper we consider a space П of centered m-planes (a space of all centered planes of the dimension m). This space is considered in the projective space n P . For the space П we have: dim П=n + (n – m)m. Principal fiber bundle is arised above it. The Lie group is a typical fiber of the principal fiber. This group acts in the tangent space to the П. Analogue of Neifeld’s connection with multivariate glueing is given in this fibering by Laptev — Lumiste way. The case when one-index forms are basic-fibre forms is considered. We realize an analogue of the Norden strong normalization of the space П by fields of the geometrical images: (n – m – 1)-plane which is not having the common points with a centered m-plane and (m – 1)-plane which is belonging to the m-plane and not passing through its centre. It is proved that the analog of the Norden strong normalization of the space of centered planes induces this connection.

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Demirel, Oğuzhan, Leyla Aslan, and Damla TOPAL. "A New Proof of the Lester’s Perimeter Theorem in Euclidean Space." Mathematical Journal of Interdisciplinary Sciences 8, no. 2 (2020): 57–59. http://dx.doi.org/10.15415/mjis.2020.82007.

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An injection defined from Euclidean space n-space E^n to itself which preserves the triangles of perimeter 1 is an Eucldean motion. J. Lester gave two different proofs for this theorem in Euclidean plane [1] and Euclidean space [2]. In this study a new technique is developed for the proof of this theorem which is valid in both Euclidean plane and Euclidean space.

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