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Journal articles on the topic 'Helicoidal surfaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 February 2022

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1

Maria da Silva, Rosângela, and Keti Tenenblat. "Helicoidal Minimal Surfaces in a Finsler Space of Randers Type." Canadian Mathematical Bulletin 57, no. 4 (December 1, 2014): 765–79. http://dx.doi.org/10.4153/cmb-2013-047-7.

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AbstractWe consider the Finsler space obtained by perturbing the Euclidean metric of ℝ3 by a rotation. It is the open region of ℝ3 bounded by a cylinder with a Randers metric. Using the Busemann–Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in . We prove that the helicoid is a minimal surface in only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space , the only minimal surfaces in the Bonnet family with fixed axis Ox̄3 are the catenoids and the helicoids.

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2

JI, FENGHUI, and YOUNG HO KIM. "HELICOIDAL CDPC-SURFACES IN MINKOWSKI 3-SPACE." International Journal of Geometric Methods in Modern Physics 07, no. 06 (September 2010): 979–88. http://dx.doi.org/10.1142/s021988781000466x.

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In this paper, we discuss the local existence and the construction of helicoidal CDPC-surfaces, i.e. the helicoidal surfaces with constant difference of the principal curvatures, in Minkowski 3-space [Formula: see text]. In general, there exist helicoidal surfaces with conjugate complex principal curvatures, which have no counterparts in R3. We prove this is still true for helicoidal CDPC-surfaces but not true for rotation surfaces in [Formula: see text].

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3

YOON, DAE WON, DONG-SOO KIM, YOUNG HO KIM, and JAE WON LEE. "HELICOIDAL SURFACES WITH PRESCRIBED CURVATURES IN Nil3." International Journal of Mathematics 24, no. 14 (December 2013): 1350107. http://dx.doi.org/10.1142/s0129167x13501073.

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In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil3. Also, we construct helicoidal surfaces in Nil3 with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

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4

Roussos, Ioannis M. "The helicoidal surfaces as Bonnet surfaces." Tohoku Mathematical Journal 40, no. 3 (1988): 485–90. http://dx.doi.org/10.2748/tmj/1178227989.

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5

Pleşu, Gheorghe. "Grapho-Analytical Determination of the Profile Disc Tool for Manufacturing of Complex Helicoidal Surfaces." Advanced Materials Research 837 (November 2013): 164–69. http://dx.doi.org/10.4028/www.scientific.net/amr.837.164.

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Helicoidal surfaces correspond to a cathegory of widely spread surfacesin the technical field. From the point of view of the manufacturing of helicoidal surfaces through the moving of a profile around the helix, the two elements defining such a surface can be distinguished: the profile and the helix. The numerical determination of the profile of the disc type tool drawn on an analytical method with a large number of points [1- (the number reaches even 1500 in some cases) implies a great effort so that the technician can verify the correctness of results, even in the present case when there is the possibility of the numerical determination through the resolution of the inverse problem of the frontal profile of the complex helicoidal surface. In order to facilitate this work and to give the possibility of introducing some new profiles in the projection system, some methods of grapho-analytical determination have been conceived to process the profiles of the complex helicoidal surfaces, as well as the profiles of the helicoidal surfaces for a given tool profile. This paper presents the graphic-analytical determination of the disc tool profile designed to dress the complex helicoidal surfaces. The methods referred to by using the possibilities provided by the programming environment AutoCAD and its development languages allow to determin numerically the profile of the disc-cutter with a high precision, above the precision level required in practice.

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6

Krivoshapko, S. N. "Geometry and Strength of General Helicoidal Shells." Applied Mechanics Reviews 52, no. 5 (May 1, 1999): 161–75. http://dx.doi.org/10.1115/1.3098932.

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The geometry of helical surfaces has been well studied. Several available methods of strength analysis of helicoidal shells give one a choice in solving one-dimensional or two-dimensional problems. The basic problems considered in this review article, which contains 181 references, include geometrical research, approximation and bending of helical surfaces, static analysis of helicoidal shells by analytic and numerical methods, the vibrations of pre-twisted cantilevered plates, helical tubular shells, the generation of helical surfaces by mated surface of revolution, and the application of the helicoidal constructions.

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7

Karunakaran, K. P., and S. G. Dhande. "Computer aided design of cutters for helicoidal surfaces." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 212, no. 5 (May 1, 1998): 373–82. http://dx.doi.org/10.1243/0954405981515978.

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The design of cutters is an important consideration for the manufacture of helicoidal surfaces such as extruder screw surfaces. These surfaces are produced mostly by milling processes using form cutters of end mill type, side mill type or disc type, such as side-and-face mill or grinding wheel. The methodology proposed in the paper addresses the problem of the design of cutters for the machining of helicoidal surfaces. Using the proposed methodology, the characteristic profile(s) of the cutter can be determined from the given cross-sectional profile and lead of a helicoidal surface. By sweeping this characteristic profile along an appropriate path or around an axis, the geometry of the specific form cutter can be obtained. Such a geometry could be a turning tool, an end mill, a side mill, a side-and-face mill or a grinding wheel, depending on the process adopted for manufacture. The proposed methodology can also be used to determine the geometry of the helicoidal surface that will be obtained by using a given cutter. In the paper, the procedure to obtain the geometry of the cutters for machining extruder screws is explained with illustrations.

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8

Jacob, Michael. "Saddle, Tower and Helicoidal Surfaces." Journal de Physique II 7, no. 7 (July 1997): 1035–44. http://dx.doi.org/10.1051/jp2:1997169.

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9

Ferrer, Leonor, and Francisco Martín. "Minimal surfaces with helicoidal ends." Mathematische Zeitschrift 250, no. 4 (April 15, 2005): 807–39. http://dx.doi.org/10.1007/s00209-005-0777-x.

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10

Волков, А. Г., Т. А. Ноговицына, А. А. Повзнер, and Т. М. Нуретдинов. "Магнитный фазовый переход в MnSi на основе LSDA+U +SO-расчетов электронной структуры и спин-флуктуационной теории." Физика твердого тела 60, no. 10 (2018): 1882. http://dx.doi.org/10.21883/ftt.2018.10.46512.105.

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AbstractSpin states appearing near the magnetic phase transition in helicoidal ferromagnet MnSi are studied on the basis of the spin-fluctuation theory and the LSDA + U + SO calculations of the electronic structure. The temperature dependence of the uniform magnetic susceptibility is calculated near the magnetic phase transition temperature, and the result agrees well with the experiment. Spin correlators corresponding to various solutions of the equation of magnetic state are determined in the region of the magnetic phase transition expanded in temperature. It is shown that, in this region, a helicoidal short-range order appears in the form of the superposition of left and right spin spirals with stochastic weight coefficients. It is shown that the magnetic susceptibility divergences on the helicoid wave vector at the temperature of disappearance of local magnetization and during the transition to the paramagnetic state.

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11

Ripoll, Jaime B. "Helicoidal minimal surfaces in hyperbolic space." Nagoya Mathematical Journal 114 (June 1989): 65–75. http://dx.doi.org/10.1017/s0027763000001409.

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Denote by H3 the 3-dimensional hyperbolic space with sectional curvatures equal to – 1, and let g be a geodesic in H3 Let {ψt} be the translation along g (see § 2) and let {φt} be the one-parameter subgroup of isometries of H3 whose orbits are circles centered on g. Given any α ∊ R, one can show that λ = {λt} = ψt ∘ φαt} is a one-parameter subgroup of isometries of H3 (see § 2) which is called a helicoidal group of isometries with angular pitch α. Any surface in H3 which is λ-invariant is called a helicoidal surface.In this work we prove some results concerning minimal helicoidal surfaces in H3. The first one reads:

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12

Chabanon, Morgan, and Padmini Rangamani. "Geometric coupling of helicoidal ramps and curvature-inducing proteins in organelle membranes." Journal of The Royal Society Interface 16, no. 158 (September 2019): 20190354. http://dx.doi.org/10.1098/rsif.2019.0354.

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Cellular membranes display an incredibly diverse range of shapes, both in the plasma membrane and at membrane bound organelles. These morphologies are intricately related to cellular functions, enabling and regulating fundamental membrane processes. However, the biophysical mechanisms at the origin of these complex geometries are not fully understood from the standpoint of membrane–protein coupling. In this study, we focused on a minimal model of helicoidal ramps representative of specialized endoplasmic reticulum compartments. Given a helicoidal membrane geometry, we asked what is the distribution of spontaneous curvature required to maintain this shape at mechanical equilibrium? Based on the Helfrich energy of elastic membranes with spontaneous curvature, we derived the shape equation for minimal surfaces, and applied it to helicoids. We showed the existence of switches in the sign of the spontaneous curvature associated with geometric variations of the membrane structures. Furthermore, for a prescribed gradient of spontaneous curvature along the exterior boundaries, we identified configurations of the helicoidal ramps that are confined between two infinitely large energy barriers. Overall our results suggest possible mechanisms for geometric control of helicoidal ramps in membrane organelles based on curvature-inducing proteins.

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13

Караштин, Е. А. "Инжекция неравновесного спина в геликоидальный ферромагнетик." Физика твердого тела 62, no. 9 (2020): 1482. http://dx.doi.org/10.21883/ftt.2020.09.49773.05h.

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The properties of spin injection into a helicoidal ferromagnet are studied. Two possible ways of spin injection are investigated: injection of spin-polarized electric current and effect of spin pumping. In the case when helicoid axis is perpendicular to the boundary through which the spin is injected the conditions of spin injection into a pre-defined spin band are determined. In the case when helicoid axis is parallel to the boundary the appearance of effect similar to topological Hall effect is shown. In the latter geometry, spin pumping leads to the exchange conversion of spin current into electric current that flows parallel to the helicoid axis.

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14

KIM, YOUNG WOOK, SUNG-EUN KOH, HEAYONG SHIN, and SEONG-DEOG YANG. "HELICOIDAL MINIMAL SURFACES IN ℍ2×ℝ." Bulletin of the Australian Mathematical Society 86, no. 1 (December 15, 2011): 135–49. http://dx.doi.org/10.1017/s0004972711003042.

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AbstractIt is shown that a minimal surface in ℍ2×ℝ is invariant under a one-parameter group of screw motions if and only if it lies in the associate family of helicoids. It is also shown that the conjugate surfaces of the parabolic and hyperbolic helicoids in ℍ2×ℝ are certain types of catenoids.

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15

Babaarslan, Murat, and Yusuf Yayli. "Differential Equation of the Loxodrome on a Helicoidal Surface." Journal of Navigation 68, no. 5 (April 27, 2015): 962–70. http://dx.doi.org/10.1017/s0373463315000181.

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In nature, science and engineering, we often come across helicoidal surfaces. A curve on a helicoidal surface in Euclidean 3-space is called a loxodrome if the curve intersects all meridians at a constant azimuth angle. Thus loxodromes are important in navigation. In this paper, we find the differential equation of the loxodrome on a helicoidal surface in Euclidean 3-space. Also we give some examples and draw the corresponding pictures via the Mathematica computer program to aid understanding of the mathematics of navigation.

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16

Bracken, Paul. "Helicoidal Surfaces and Their Relationship to Bonnet Surfaces." Advances in Pure Mathematics 07, no. 01 (2017): 31–40. http://dx.doi.org/10.4236/apm.2017.71003.

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17

Romon, Pascal. "On helicoidal ends of minimal surfaces." Annals of Global Analysis and Geometry 12, no. 1 (February 1994): 341–55. http://dx.doi.org/10.1007/bf02108306.

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18

Traizet, Martin, and Matthias Weber. "Hermite polynomials and helicoidal minimal surfaces." Inventiones mathematicae 161, no. 1 (February 28, 2005): 113–49. http://dx.doi.org/10.1007/s00222-004-0420-1.

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19

Hoffman, David, Martin Traizet, and Brian White. "Helicoidal minimal surfaces of prescribed genus." Acta Mathematica 216, no. 2 (2016): 217–323. http://dx.doi.org/10.1007/s11511-016-0139-z.

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20

Повзнер, А. А., Т. М. Нуретдинов, and А. Г. Волков. "Исследование квантовых флуктуаций в Fe-=SUB=-x-=/SUB=-Mn-=SUB=-1-x-=/SUB=-Si с учетом LDA + U + SO-расчетов электронной структуры." Физика твердого тела 61, no. 4 (2019): 630. http://dx.doi.org/10.21883/ftt.2019.04.47404.307.

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AbstractQuantum spin fluctuations are studied in chiral helicoidal Fe_ x Mn_1 – _ x Si ferromagnets on the basis of the direct calculations of the electronic structure. The disappearance of the helicoidal long-range order is shown to be accompanied by the appearance of the crossover of the thermodynamic and quantum transitions with a sharp decrease in the local magnetization and the zero fluctuation amplitude. Suppression of the local magnetization by quantum spin fluctuations leads to the concentration–temperature magnetic transition with the disappearance of the helicoidal short-range order. The calculations of the magnetization of the Fe_ x Mn_1 – _ x Si compositions show that at 0.12 < x < 0.20 the magnetic state in the thermodynamic limit is characterized by the short-range (not long-range) helicoidal order.

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21

Yıldız, Önder. "Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density." Mathematics 7, no. 1 (December 28, 2018): 27. http://dx.doi.org/10.3390/math7010027.

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In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We get a result for the minimal case. Additionally, we give examples of a helicoidal surface with a weighted mean curvature and weighted extrinsic curvature.

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22

Yoon, Dae, and Jae Lee. "Linear Weingarten Helicoidal Surfaces in Isotropic Space." Symmetry 8, no. 11 (November 14, 2016): 126. http://dx.doi.org/10.3390/sym8110126.

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23

Beneki, Chr C., G. Kaimakamis, and B. J. Papantoniou. "Helicoidal surfaces in three-dimensional Minkowski space." Journal of Mathematical Analysis and Applications 275, no. 2 (November 2002): 586–614. http://dx.doi.org/10.1016/s0022-247x(02)00269-x.

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24

Perdomo, Oscar M. "Helicoidal minimal surfaces in $\mathbf{R}^{3}$." Illinois Journal of Mathematics 57, no. 1 (2013): 87–104. http://dx.doi.org/10.1215/ijm/1403534487.

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25

Manfio, Fernando, and João Paulo dos Santos. "Helicoidal flat surfaces in the 3-sphere." Mathematische Nachrichten 292, no. 1 (August 14, 2018): 127–36. http://dx.doi.org/10.1002/mana.201700254.

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26

Kuhns, Chad, and Bennett Palmer. "Helicoidal surfaces with constant anisotropic mean curvature." Journal of Mathematical Physics 52, no. 7 (July 2011): 073506. http://dx.doi.org/10.1063/1.3603816.

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27

Mira, Pablo, and Jos� A. Pastor. "Helicoidal Maximal Surfaces in Lorentz-Minkowski Space." Monatshefte f�r Mathematik 140, no. 4 (November 1, 2003): 315–34. http://dx.doi.org/10.1007/s00605-003-0111-9.

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28

Martínez, Antonio, João dos Santos, and Keti Tenenblat. "Helicoidal flat surfaces in hyperbolic 3-space." Pacific Journal of Mathematics 264, no. 1 (July 5, 2013): 195–211. http://dx.doi.org/10.2140/pjm.2013.264.195.

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29

Yıldız, Önder Gökmen, Selman Hızal, and Mahmut Akyiğit. "Type I+ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 3 (December 1, 2018): 99–108. http://dx.doi.org/10.2478/auom-2018-0035.

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AbstractIn this paper, we construct a helicoidal surface of type I+ with prescribed weighted mean curvature and Gaussian curvature in the Minkowski 3−space ${\Bbb R}_1^3$with a positive density function. We get a result for minimal case. Also, we give examples of a helicoidal surface with weighted mean curvature and Gaussian curvature.

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30

Karacan, Murat Kemal, Dae Won Yoon, and Sezai Kiziltug. "Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹." Tamkang Journal of Mathematics 48, no. 2 (June 30, 2017): 123–34. http://dx.doi.org/10.5556/j.tkjm.48.2017.2200.

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In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space I₃¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

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31

Choi, Mie-Kyung, Dong-Soo Kim, and Young-Ho Kim. "HELICOIDAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP." Journal of the Korean Mathematical Society 46, no. 1 (January 31, 2009): 215–23. http://dx.doi.org/10.4134/jkms.2009.46.1.215.

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32

Baikoussis, Christos, and Themis Koufogiorgos. "Helicoidal surfaces with prescribed mean or Gaussian curvature." Journal of Geometry 63, no. 1-2 (November 1998): 25–29. http://dx.doi.org/10.1007/bf01221235.

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33

Lisitsa, V. T. "n-Dimensional helicoidal surfaces in Euclidean space Em." Mathematical Notes of the Academy of Sciences of the USSR 41, no. 4 (April 1987): 308–12. http://dx.doi.org/10.1007/bf01137680.

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34

SASAI, Takao. "On Helicoidal Surfaces with Constant Mean Curvature and Their Limiting Surfaces." Tokyo Journal of Mathematics 19, no. 1 (June 1996): 39–50. http://dx.doi.org/10.3836/tjm/1270043217.

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35

Ji, Fenghui, and Young Ho Kim. "Isometries between minimal helicoidal surfaces and rotation surfaces in minkowski space." Applied Mathematics and Computation 220 (September 2013): 1–11. http://dx.doi.org/10.1016/j.amc.2013.05.052.

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36

Babaarslan, Murat, and Mustafa Kayacik. "Time-like loxodromes on helicoidal surfaces in Minkowski 3-space." Filomat 31, no. 14 (2017): 4405–14. http://dx.doi.org/10.2298/fil1714405b.

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37

Ersoy, Soley, and Kemal Eren. "Timelike Tangent Developable Surfaces and Bonnet Surfaces." Abstract and Applied Analysis 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/6837543.

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A criterion was given for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space by Chen and Li, 1999. In this study, we obtain a necessary and sufficient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the aid of this criterion. This is examined under the condition of the curvature and torsion of the base curve of the timelike developable surface being nonconstant. Moreover, we investigate the nontrivial isometry preserving the mean curvature for a timelike flat helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.

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38

Araujo, Kellcio Oliveira, Ningwei Cui, and Romildo da Silva Pina. "HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE." Bulletin of the Korean Mathematical Society 53, no. 2 (March 31, 2016): 531–40. http://dx.doi.org/10.4134/bkms.2016.53.2.531.

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39

Yoon, Dae, Dong-Soo Kim, Young Kim, and Jae Lee. "Constructions of Helicoidal Surfaces in Euclidean Space with Density." Symmetry 9, no. 9 (August 28, 2017): 173. http://dx.doi.org/10.3390/sym9090173.

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40

Halldorsson, Hoeskuldur P. "Helicoidal surfaces rotating/translating under the mean curvature flow." Geometriae Dedicata 162, no. 1 (March 17, 2012): 45–65. http://dx.doi.org/10.1007/s10711-012-9716-2.

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41

Hou, Zhong Hua, and Fenghui Ji. "Helicoidal surfaces with H2=K in Minkowski 3-space." Journal of Mathematical Analysis and Applications 325, no. 1 (January 2007): 101–13. http://dx.doi.org/10.1016/j.jmaa.2006.01.017.

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42

Повзнер, А. А., А. Г. Волков, and Т. М. Нуретдинов. "Концентрационные флуктуации в киральных ферромагнетиках Fe-=SUB=-x-=/SUB=-Mn-=SUB=-1-x-=/SUB=-Si во внешнем магнитном поле." Физика твердого тела 62, no. 5 (2020): 776. http://dx.doi.org/10.21883/ftt.2020.05.49246.648.

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In the framework of the theory of spin fluctuations, magnetic h-T diagrams of chiral helicoidal ferromagnets FexMn1-xSi with the Dzyaloshinskii-Moriya interaction are investigated. A specific analysis of the equations of magnetic state is carried out on the basis of a model of the electronic structure following from the LDA + U + SO DOS calculations in the virtual crystal approximation. It was shown that in the concentration range x <0.12, the Fermi level remains within the local minimum DOS. In this case, a helicoidal long-range order is realized, which undergoes a first-order transition induced by spin fluctuations, accompanied by the formation of intermediate skyrmion phases induced by an external magnetic field. With increasing x arising due to the chaotic distribution of the magnetic moments of manganese and iron over the nodes, the effects of concentration fluctuations suppress zero-point quantum spin fluctuations. In this case, the condition for the appearance of skyrmion phases is violated for x> 0.12, and the region of the helicoidal ferromagnetic order is preserved up to concentrations xc = 0.20. In the interval 0.10 <x <0.20, the transition induced by fluctuations to the paramagnetic state is accompanied by the disappearance of local magnetization and the formation of a paramagnetic state with dynamic spin correlations.

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43

Choi, Mie-Kyung, Young-Ho Kim, Huili Liu, and Dae-Won Yoon. "HELICOIDAL SURFACES AND THEIR GAUSS MAP IN MINKOWSKI 3-SPACE." Bulletin of the Korean Mathematical Society 47, no. 4 (July 31, 2010): 859–81. http://dx.doi.org/10.4134/bkms.2010.47.4.859.

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44

Hoffman, David, and Brian White. "Axial minimal surfaces in $S^2 x R$ are helicoidal." Journal of Differential Geometry 87, no. 3 (March 2011): 515–24. http://dx.doi.org/10.4310/jdg/1312998234.

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45

Schief, W. K., and C. Rogers. "Modulated waves and helicoidal pseudospherical surfaces in nonlinear inhomogeneous elasticity." Journal of Physics A: Mathematical and Theoretical 43, no. 10 (February 22, 2010): 105206. http://dx.doi.org/10.1088/1751-8113/43/10/105206.

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46

Ji, Fenghui, and Zhong Hua Hou. "A kind of helicoidal surfaces in 3-dimensional Minkowski space." Journal of Mathematical Analysis and Applications 304, no. 2 (April 2005): 632–43. http://dx.doi.org/10.1016/j.jmaa.2004.09.065.

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47

Повзнер, А. А., А. Г. Волков, and Т. А. Ноговицына. "Электронная структура и магнитный фазовый переход в геликоидальных ферромагнетиках Fe-=SUB=-1-x-=/SUB=-Co-=SUB=-x-=/SUB=-Si." Физика твердого тела 60, no. 2 (2018): 227. http://dx.doi.org/10.21883/ftt.2018.02.45372.232.

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AbstractLSDA + U + SO calculations of the electronic structure of helicoidal Fe_1 - x Co_ x Si ferromagnets within the virtual crystal approximation have been supplemented with the consideration of the Dzyaloshinski-Moriya interaction and ferromagnetic fluctuations of the spin density of collective d electrons with the Hubbard interactions at Fe and Co atoms randomly distributed over sites. The magnetic-state equation in the developed model describes helicoidal ferromagnetism and its disappearance accompanied by the occurrence of a maximum of uniform magnetic susceptibility at temperature T _ C and chiral fluctuations of the local magnetization at T > T _ C . The reasons why the magnetic contribution to the specific heat at the magnetic phase transition changes monotonically and the volume coefficient of thermal expansion (VCTE) at low temperatures is negative and has a wide minimum near T _ C have been investigated. It is shown that the VCTE changes sign when passing to the paramagnetic state (at temperature T _ S ).

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48

Radzevitch, S. P., and E. D. Goodman. "About the Orthogonal Parameterization of Sculptured Part Surfaces and Initial Tool Surfaces." Journal of Manufacturing Science and Engineering 119, no. 4B (November 1, 1997): 823–28. http://dx.doi.org/10.1115/1.2836830.

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In the domain of multi-axis NC machining of sculptured surface parts, the use of orthogonal parameterizations of part and tool surfaces is convenient because it simplifies the transformation of coordinate systems. Using the so-called “differential-geometric method of sculptured surface NC machining,” developed by one of the authors, many parameterizations of part and tool surfaces are easily shown not to be orthogonal. To transform nonorthogonal part and tool surface parameterizations into orthogonal ones, the Jacobian of the transformation may be used. In cases when the Jacobian of the transformation is not known, it is possible to use differential equations for isogonal trajectories on the surfaces (choosing an orthogonal case), or a special kinematic method for obtaining sculptured surface equations. Influences of coordinate system transformations (translations and rotations along and about axes through the origin) on example part and tool surface parameterizations for four types of general helicoidal surfaces are described. The results mentioned above simplify the analytical description of the multi-axis NC machining process, and may be useful for writing NC toolpath generation software.

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SASAHARA, Naoshi. "Spacelike Helicoidal Surfaces with Constant Mean Curvature in Minkowski 3-space." Tokyo Journal of Mathematics 23, no. 2 (December 2000): 477–502. http://dx.doi.org/10.3836/tjm/1255958684.

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Choi, Mie-Kyung, Young-Ho Kim, and Gi-Chan Park. "HELICOIDAL SURFACES AND THEIR GAUSS MAP IN MINKOWSKI 3-SPACE II." Bulletin of the Korean Mathematical Society 46, no. 3 (May 31, 2009): 567–76. http://dx.doi.org/10.4134/bkms.2009.46.3.567.

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