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Journal articles on the topic 'Normal curves'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Haggui, Fathi, and Adel Khalfallah. "Normal pseudoholomorphic curves." Annales Polonici Mathematici 101, no. 1 (2011): 55–65. http://dx.doi.org/10.4064/ap101-1-6.

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2

KÖRPINAR, Talat, and Ahmet SAZAK. "E3'te Adjoint Eğriler ile İlişkili Normal Yüzeyler." Bitlis Eren Üniversitesi Fen Bilimleri Dergisi 11, no. 3 (2022): 779–83. http://dx.doi.org/10.17798/bitlisfen.1086972.

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In this study, we work on the surfaces determined in relation to associated curves. We study normal surfaces defined with the help of adjoint curves, a special type of associated curve. For this, we first remember the basic equations of the 3-dimensional Euclidean space, which is the space we work with, and the adjoint curve issue. Then, by computing the first and second fundamental forms, principal curvatures and Gaussian and mean curvatures of the normal surface of an adjoint curve, we obtain the characterizations of this surface and related some results.

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3

ECHABBI, Nidal, and Amina OUAZZANI CHAHDI. "Some Associated Curves of Normal Indicatrix of a Regular Curve." Journal of Mathematics Research 12, no. 1 (2020): 84. http://dx.doi.org/10.5539/jmr.v12n1p84.

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In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.

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4

ECHABBI, Nidal, and Amina OUAZZANI CHAHDI. "Some Associated Curves of Normal Indicatrix of a Regular Curve." Journal of Mathematics Research 12, no. 1 (2020): 92. http://dx.doi.org/10.5539/jmr.v12n1p92.

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In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.

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5

Çelik, Oğuzhan. "A Generalization of Curve Mates: Normal Mate of a Curve." Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 17, no. 2 (2024): 338–52. http://dx.doi.org/10.18185/erzifbed.1396745.

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This a paper, a new curve pair is defined that generalizes some pairs of curves well known as Mannheim and Bertrand curve pairs. A normal curve pair is defined in such a way that a vector u obtained by overlapping the normal planes of the G and G* curves makes the same angle as the binormals of these curves. The relationship between torsions and curvatures of curve pairs was analyzed. Moreover, The unit quaternion q corresponding to the rotation matrix between the Frenet vectors of the curves was defined. In the conclusion, it is expressed express which famous pairs of curves will be obtained

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6

Fløystad, Gunnar. "Curves on normal surfaces." Transactions of the American Mathematical Society 352, no. 12 (2000): 5485–510. http://dx.doi.org/10.1090/s0002-9947-00-02054-7.

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7

Levcovitz, Daniel, and Timothy C. McCune. "Projectively normal involutive curves." Journal of Pure and Applied Algebra 174, no. 2 (2002): 153–62. http://dx.doi.org/10.1016/s0022-4049(02)00045-2.

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8

Alghanemi, Azeb, and Abeer AlGhawazi. "Some Geometric Characterizations of f -Curves Associated with a Plane Curve via Vector Fields." Advances in Mathematical Physics 2022 (April 27, 2022): 1–9. http://dx.doi.org/10.1155/2022/9881237.

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The differential geometry of plane curves has many applications in physics especially in mechanics. The curvature of a plane curve plays a role in the centripetal acceleration and the centripetal force of a particle traversing a curved path in a plane. In this paper, we introduce the concept of the f -curves associated with a plane curve which are more general than the well-known curves such as involute, evolute, parallel, symmetry set, and midlocus. In fact, we introduce the f -curves associated with a plane curve via its normal and tangent for both the cases, a Frenet curve and a Legendre cu

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9

Erdem, Hatice Altın, and Kazım İlarslan. "Spacelike Bertrand curves in Minkowski 3-space revisited." Analele Universitatii "Ovidius" Constanta - Seria Matematica 31, no. 3 (2023): 87–109. https://doi.org/10.2478/auom-2023-0033.

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Abstract In the geometry of curves in 𝔼3, if the principal normal vector field of a given space curve ϕ with non-zero curvatures is the principal normal vector field of another space curve ϕ *, then the curve ϕ is called a Bertrand curve and ϕ * is called Bertrand partner of ϕ. These curves have been studied in di erent space over a long period of time and found wide application in di erent areas. Therefore, we have a great knowledge of geometric properties of these curves. In this paper, revested results for spacelike Bertrand curves with non-null normal vectors will be given with the previou

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10

Ballico, E., and A. Cossidente. "Osculating subspaces, normal rational curves and generalized strange curves." Bulletin of the Belgian Mathematical Society - Simon Stevin 8, no. 1 (2001): 87–94. http://dx.doi.org/10.36045/bbms/1102714031.

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11

Hananoi, Satoshi, and Shyuichi Izumiya. "Normal developable surfaces of surfaces along curves." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 1 (2017): 177–203. http://dx.doi.org/10.1017/s030821051600007x.

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We consider a developable surface normal to a surface along a curve on the surface. We call it a normal developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface that characterize these singularities.

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12

Kiziltug, Sezai, Mehmet Önder, and Yusuf Yayli. "Normal direction curves and applications." Miskolc Mathematical Notes 22, no. 1 (2021): 363. http://dx.doi.org/10.18514/mmn.2021.1476.

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13

Dolcetti, A., and G. Pareschi. "On linearly normal space curves." Mathematische Zeitschrift 198, no. 1 (1988): 73–82. http://dx.doi.org/10.1007/bf01183040.

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14

Kremer, Karsten. "Normal origamis of Mumford curves." manuscripta mathematica 133, no. 1-2 (2010): 83–103. http://dx.doi.org/10.1007/s00229-010-0379-8.

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15

Gmainer, Johannes, and Hans Havlicek. "Nuclei of normal rational curves." Journal of Geometry 69, no. 1-2 (2000): 117–30. http://dx.doi.org/10.1007/bf01237480.

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16

Runborg, Olof, Wim Sweldens, and Ingrid Daubechies. "Normal Multiresolution Approximation of Curves." Constructive Approximation 20, no. 3 (2004): 399–463. http://dx.doi.org/10.1007/s00365-003-0543-4.

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17

ŞENYURT, Süleyman, Davut CANLI, and Kebire Hilal AYVACI. "Associated curves from a different point of view in $E^3$." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 3 (2022): 826–45. http://dx.doi.org/10.31801/cfsuasmas.1026359.

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In this paper, tangent, principal normal and binormal wise associated curves are defined such that each of these vectors of any given curve lies on the osculating, normal and rectifying plane of its partner, respectively. For each associated curve, a new moving frame and the corresponding curvatures are formulated in terms of Frenet frame vectors. In addition to this, the possible solutions for distance functions between the curve and its associated mate are discussed. In particular, it is seen that the involute curves belong to the family of tangent associated curves in general and the Bertra

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18

Nesovic, Emilija, Ufuk Öztürk, and Öztürk Koç. "On non-null relatively normal-slant helices in Minkowski 3-space." Filomat 36, no. 6 (2022): 2051–62. http://dx.doi.org/10.2298/fil2206051n.

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By using the Darboux frame |?, ?, ?| of a non-null curve lying on a timelike surface in Minkowski 3-space, where ? is the unit tangent vector of the curve, ? is the unit spacelike normal vector field restricted to the curve and ? = ?? ? ?, we define relatively normal-slant helices as the curves satisfying the condition that the scalar product of the fixed vector spanning their axis and the non-constant vector field ? is constant. We give the necessary and sufficient conditions for non-null curves lying on a timelike surface to be relatively normal-slant helices. We consider the special cases w

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19

Olejníková, Tatiana. "Cycloids on the Spherical Surface." Selected Scientific Papers - Journal of Civil Engineering 10, no. 1 (2015): 27–36. http://dx.doi.org/10.1515/sspjce-2015-0003.

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Abstract The article describes the creation of the normal cycloidal curve by rotation of the point about the normal of the circle on the spherical surface and binormal cycloidal curve created by rotation of the point about the binormal of the previous cycloidal curve. Cycloidal cyclical surfaces are created by moving circles along the curves lying in the normal plane of the curves. Described cycloidal cyclical surfaces are projected on the spherical surface. Varying parameters of the curves are generated different ornaments on the spherical surface.

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20

Chen, Jing, and Zhoudao Lu. "Crack Extension Resistance of Normal-Strength Concrete Subjected to Elevated Temperatures." Advances in Materials Science and Engineering 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/683756.

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Determination of the residual crack extension resistance curves (KR-curves) associated with cohesive force distribution on fictitious crack zone of complete fracture process is implemented in present research. The cohesive force distributes according to bilinear softening traction-separation law proposed by Petersson. Totally ten temperatures varying from 20°C to 600°C and the specimen size of230×200×200 mm with initial-notch depth ratios 0.4 are considered. The load-crack mouth opening displacement curves (P-CMOD) of postfire specimens are obtained by wedge-splitting method from which the str

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21

Qian, Jinhua, Xueqian Tian, and Young Ho Kim. "Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms." Mathematics 8, no. 6 (2020): 919. http://dx.doi.org/10.3390/math8060919.

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In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme.

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22

Zainal, Abdul Kareem Esmat. "Mathematical Modeling of Compaction Curve Using Normal Distribution Functions." Journal of Engineering 24, no. 2 (2018): 118–30. http://dx.doi.org/10.31026/j.eng.2018.02.08.

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Compaction curves are widely used in civil engineering especially for road constructions, embankments, etc. Obtaining the precise amount of Optimum Moisture Content (OMC) that gives the Maximum Dry Unit weight gdmax. is very important, where the desired soil strength can be achieved in addition to economic aspects. In this paper, three peak functions were used to obtain the OMC and gdmax. through curve fitting for the values obtained from Standard Proctor Test. Another surface fitting was also used to model the Ohio’s compaction curves that represent the very large variation of compacted

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23

Koshkin, Sergiy, and Ivan Rocha. "Caustics of Light Rays and Euler's Angle of Inclination." PUMP Journal of Undergraduate Research 3 (October 22, 2020): 205–25. http://dx.doi.org/10.46787/pump.v3i0.2417.

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Euler used intrinsic equations expressing the radius of curvature as a function of the angle of inclination to find curves similar to their evolutes. We interpret the evolute of a plane curve optically, as the caustic (envelope) of light rays normal to it, and study the Euler's problem for general caustics. The resulting curves are characterized when the rays are at a constant angle to the curve, generalizing the case of evolutes. Aside from analogs of classical solutions we encounter some new types of curves. We also consider caustics of parallel rays reflected by a curved mirror, where Euler

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24

Duyaguit, Ma Cristina Lumakin, and Hisao Yoshihara. "Galois Lines for Normal Elliptic Space Curves." Algebra Colloquium 12, no. 02 (2005): 205–12. http://dx.doi.org/10.1142/s1005386705000192.

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Let C be a curve, and l and l0 be lines in the projective three space ℙ3. Consider a projection πl: ℙ3 ⋯ → l0 with center l, where l ⋂ l0= ∅. Restricting πl to C, we obtain a morphism πl|C : C → l0 and an extension of fields (πl|C)* : k(l0) ↪ k(C). If this extension is Galois, then l is said to be a Galois line. We study the defining equations, automorphisms and the Galois lines for quartic curves, and give some applications to the theory of plane quartic curves.

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25

Hanif, Muhammad, Zhong Hou, and Kottakkaran Nisar. "On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space." Symmetry 10, no. 8 (2018): 317. http://dx.doi.org/10.3390/sym10080317.

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Recently, extensive research has been done on evolute curves in Minkowski space-time. However, the special characteristics of curves demand advanced level observations that are lacking in existing well-known literature. In this study, a special kind of generalized evolute and involute curve is considered in four-dimensional Minkowski space. We consider (1,3)-evolute curves with respect to the casual characteristics of the (1,3)-normal plane that are spanned by the principal normal and the second binormal of the vector fields and the (0,2)-evolute curve that is spanned by the tangent and first

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26

O'Donnell, C. R., R. G. Castile, and J. Mead. "Changes in flow-volume curve configuration with bronchoconstriction and bronchodilation." Journal of Applied Physiology 61, no. 6 (1986): 2243–51. http://dx.doi.org/10.1152/jappl.1986.61.6.2243.

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Changes in the configuration of maximum expiratory flow-volume (MEFV) curves following mild degrees of bronchodilation or bronchoconstriction were studied in five normal and five asthmatic subjects. In a volume-displacement plethysmograph, MEFV curves were performed before and after inhalation of aerosolized isoproterenol (I) or histamine (H). Five filtered MEFV curves were averaged, and slope ratio vs. volume (SR-V) plots were obtained from averaged curves. Following I, maximal flows at 75% of the vital capacity (VC) were decreased in asthmatics but not in normal subjects. Flows at 50 and 25%

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27

Park, Jisoon, Taewon Kim, Seung-Yeob Baek, and Kunwoo Lee. "An algorithm for estimating surface normal from its boundary curves." Journal of Computational Design and Engineering 2, no. 1 (2014): 67–72. http://dx.doi.org/10.1016/j.jcde.2014.11.007.

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Abstract Recently, along with the improvements of geometry modeling methods using sketch-based interface, there have been a lot of developments in research about generating surface model from 3D curves. However, surfacing a 3D curve network remains an ambiguous problem due to the lack of geometric information. In this paper, we propose a new algorithm for estimating the normal vectors of the 3D curves which accord closely with user intent. Bending energy is defined by utilizing RMF(Rotation-Minimizing Frame) of 3D curve, and we estimated this minimal energy frame as the one that accords design

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28

ALTINBAŞ, Hasan. "Spacelike Ac-Slant Curves with Non-Null Principal Normal in Minkowski 3-Space." Journal of New Theory, no. 45 (December 31, 2023): 120–30. http://dx.doi.org/10.53570/jnt.1401001.

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In this paper, we define a spacelike ac-slant curve whose scalar product of its acceleration vector and a unit non-null fixed direction is a constant in Minkowski 3-space. Furthermore, we give a characterization depending on the curvatures of the spacelike ac-slant curve. After that, we get the relationship between a spacelike ac-slant curve and several distinct types of curves, such as spacelike Lorentzian spherical curves, spacelike helices, spacelike slant helices, and spacelike Salkowski curves, enhancing our understanding of its geometric properties in Minkowski 3-space. Finally, we used

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29

Sharma, Sandeep, and Kuljeet Singh. "Some Aspects of Rectifying Curves on Regular Surfaces Under Different Transformations." International Journal of Analysis and Applications 21 (July 27, 2023): 78. http://dx.doi.org/10.28924/2291-8639-21-2023-78.

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An essential space curve in the study of differential geometry is the rectifying curve. In this paper, we studied the adequate requirement for a rectifying curve under the isometry of the surfaces. The normal components of the rectifying curves are also studied, and it is investigated that for rectifying curves, the Christoffel symbols and the normal components along the surface normal are invariant under the isometric transformation. Moreover, we also studied some properties for the first fundamental form of the surfaces.

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30

Gimigliano, Alessandro. "A construction of projectively normal curves." ANNALI DELL UNIVERSITA DI FERRARA 34, no. 1 (1988): 161–81. http://dx.doi.org/10.1007/bf02824981.

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31

Ballico, E. "Hyperplane sections of linearly normal curves." Proceedings of the American Mathematical Society 122, no. 2 (1994): 395. http://dx.doi.org/10.1090/s0002-9939-1994-1213855-5.

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32

Edwards, Harold M. "A normal form for elliptic curves." Bulletin of the American Mathematical Society 44, no. 03 (2007): 393–423. http://dx.doi.org/10.1090/s0273-0979-07-01153-6.

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33

Carlini, E., and M. V. Catalisano. "Existence results for rational normal curves." Journal of the London Mathematical Society 76, no. 1 (2007): 73–86. http://dx.doi.org/10.1112/jlms/jdm042.

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34

Schröer, S. "On contractible curves on normal surfaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2000, no. 524 (2000): 1–15. http://dx.doi.org/10.1515/crll.2000.057.

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35

Fisher, Tom. "Pfaffian presentations of elliptic normal curves." Transactions of the American Mathematical Society 362, no. 5 (2009): 2525–40. http://dx.doi.org/10.1090/s0002-9947-09-04876-4.

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36

Giansiracusa, Noah. "Conformal blocks and rational normal curves." Journal of Algebraic Geometry 22, no. 4 (2013): 773–93. http://dx.doi.org/10.1090/s1056-3911-2013-00601-3.

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37

Harizanov, S., P. Oswald, and T. Shingel. "Normal Multi-scale Transforms for Curves." Foundations of Computational Mathematics 11, no. 6 (2011): 617–56. http://dx.doi.org/10.1007/s10208-011-9104-6.

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38

Ellia, Ph. "Self-linked curves and normal bundle." Journal of Pure and Applied Algebra 219, no. 1 (2015): 77–82. http://dx.doi.org/10.1016/j.jpaa.2014.04.010.

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39

Brown, S. G. R., R. W. Evans, and B. Wilshire. "Exponential descriptions of normal creep curves." Scripta Metallurgica 20, no. 6 (1986): 855–60. http://dx.doi.org/10.1016/0036-9748(86)90454-0.

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40

Brevik, John. "Curves on normal rational cubic surfaces." Pacific Journal of Mathematics 230, no. 1 (2007): 73–105. http://dx.doi.org/10.2140/pjm.2007.230.73.

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41

Vargas, J. A. "Dynamical systems for rational normal curves." Collectanea mathematica 59, no. 3 (2008): 325–46. http://dx.doi.org/10.1007/bf03191191.

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42

Catalisano, Maria Virginia, Philippe Ellia, and Alessandro Gimigliano. "Fat Points on Rational Normal Curves." Journal of Algebra 216, no. 2 (1999): 600–619. http://dx.doi.org/10.1006/jabr.1998.7761.

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43

Rafael Sendra, J. "Normal Parametrizations of Algebraic Plane Curves." Journal of Symbolic Computation 33, no. 6 (2002): 863–85. http://dx.doi.org/10.1006/jsco.2002.0538.

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44

Nielsen, Frank. "A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions." Entropy 25, no. 4 (2023): 654. http://dx.doi.org/10.3390/e25040654.

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We present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. We consider experimentally the linear interpolation curves in the ordinary, natural, and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller’s isometric embedding of the Fisher–Rao d-variate normal man

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45

Güvenç, Şaban, and Cihan Özgür. "Magnetic Curves in Homothetic s-th Sasakian Manifolds." Mathematics 13, no. 1 (2025): 159. https://doi.org/10.3390/math13010159.

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We investigate normal magnetic curves in (2n+s)-dimensional homothetic s-th Sasakian manifolds as a generalization of S-manifolds. We show that a curve γ is a normal magnetic curve in a homothetic s-th Sasakian manifold if and only if its osculating order satisfies r≤3 and it belongs to a family of θi-slant helices. Additionally, we construct a homothetic s-th Sasakian manifold using generalized D-homothetic transformations and present the parametric equations of normal magnetic curves in this manifold.

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46

Skuratovskii, Ruslan, and Volodymyr Osadchyy. "Criterions of Supersinguliarity and Groups of Montgomery and Edwards Curves in Cryptography." WSEAS TRANSACTIONS ON MATHEMATICS 19 (March 1, 2021): 709–22. http://dx.doi.org/10.37394/23206.2020.19.77.

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We consider the algebraic affine and projective curves of Edwards over the finite field Fpn. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. The criterions of the supersingularity of Montgomery and Edwards curves are found. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field and we construct birational isomorphism of them with cubic in Weierstrass normal form. One class of twisted Edwards is researched too. We propose a novel effective method of point counting for both Edwards and elliptic cur

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47

Appleboim, Eli. "From Normal Surfaces to Normal Curves to Geodesics on Surfaces." Axioms 6, no. 4 (2017): 26. http://dx.doi.org/10.3390/axioms6030026.

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48

DE POI, PIETRO, and FRANCESCO ZUCCONI. "FERMAT HYPERSURFACES AND SUBCANONICAL CURVES." International Journal of Mathematics 22, no. 12 (2011): 1763–85. http://dx.doi.org/10.1142/s0129167x11007410.

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We extend the classical Enriques–Petri theorem to s-subcanonical projectively normal curves, proving that such a curve is (s + 2)-gonal if and only if it is contained in a surface of minimal degree. We also show that any Fermat hypersurface of degree s + 2 is apolar to a s-subcanonical (s + 2)-gonal projectively normal curve, and vice versa.

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49

LIN, RUEY-MO, RONG-SHEAN LEE, YING-MING HUANG, SHOU-I. CHEN, and CHIN-YIN YU. "ANALYSIS OF LUMBOSACRAL LORDOSIS USING STANDING LATERAL RADIOGRAPHS THROUGH CURVE RECONSTRUCTION." Biomedical Engineering: Applications, Basis and Communications 14, no. 04 (2002): 149–56. http://dx.doi.org/10.4015/s101623720200022x.

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The consensus of the normal magnitude of lumbosacral curve has not been achieved. The Cobb's angle cannot depict the whole contour of this curve. For practical applications, a clearer image of these curves and their aging changes should be further investigated. This study aimed to provide a more consolidate concept of normal lumbosacral curves for clinician through a computerized reconstruction method. Standing lateral radiographs of lumbosacral spine in 82 normal adults were used for reconstructing the sagittal lumbosacral curves. The geometric characteristics of these curves according to the

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50

Li, Yanlin, Osman Keçilioğlu, and Kazım İlarslan. "Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space." Axioms 14, no. 4 (2025): 253. https://doi.org/10.3390/axioms14040253.

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In this study, the existence of Bertrand curves (in the classical sense, i.e., curves with a common principal normal vector field) in four-dimensional Euclidean space is demonstrated using a novel approach. The necessary conditions for a regular curve to be a Bertrand curve pair are obtained. Furthermore, the relationship between Bertrand curves and Combescure-related curves (pairs of curves with parallel Frenet vectors) is established, and several geometric properties are derived. Additionally, examples are constructed for both Bertrand curve pairs and Combescure-related curve pairs, and thei

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