Curvature of the Ellipsoid with Cartesian Coordinates

Sebahattin Bektas

doi:doi:10.11648/j.larp.20170202.13

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Curvature of the Ellipsoid with Cartesian Coordinates

Sebahattin Bektas

Published in Landscape Architecture and Regional Planning (Volume 2, Issue 2)

Received: 19 January 2017 Accepted: 4 February 2017 Published: 4 March 2017

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Abstract

This study aims to show how to obtain the curvature of the ellipsoid depending on azimuth angle. The curvature topic is quite popular at an interdisciplinary level. It can be to the friends of geometry, geodesy, satellite orbits in space, in studying all sorts of elliptical motions (e.g., planetary motions), curvature of surfaces and concerning eye-related radio-therapy treatment, for example the anterior surface of the cornea is often represented as ellipsoidal in form. On the calculation of the curvature, there is a famous Euler formula for rotating ellipsoid that everyone knows. Let θ be the angle, in the tangent plane, measured clockwise from the direction of minimum curvature κ₁. Then the normal curvature κ_n (θ) in direction θ is given by κ_n (θ) = κ₁ cos²θ + κ₂ sin²θ = κ₁ + (κ₂ - κ₁) cos²θ I wonder how can a formula for a triaxial ellipsoid? So we started to work. And we finally found the formula for the triaxial ellipsoid.

Published in	Landscape Architecture and Regional Planning (Volume 2, Issue 2)
DOI	10.11648/j.larp.20170202.13
Page(s)	61-66
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

General Ellipsoid, Normal Section Curve, Principal Curvatures, Gaussian Curvature, Mean Curvature

References

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[2]	Barbero, S. “The concept of geodesic curvature applied to optical surfaces”, Ophthalmic Physiol Opt 2015. doi: 10.1111/opo.12216 [1].
[3]	Bektas, S, “Orthogonal Distance From An Ellipsoid”, Boletim de Ciencias Geodesicas, Vol. 20, No. 4 ISSN 1982-2170, http://dx.doi.org/10.1590/S1982-217020140004000400053, 2014.
[4]	Bektas, S, “Least squares fitting of ellipsoid using orthogonal distances”, Boletim de Ciencias Geodesicas, Vol. 21, No. 2 ISSN 1982-2170, http://dx.doi.org/10.1590/S1982-21702015000200019, 2015a.
[5]	Bektas, S, Geodetic Computations on Triaxial Ellipsoid, International Journal of Mining Science (IJMS) Volume 1, Issue 1, June 2015b, PP 25-34 www.arcjournals.org ©ARC Page \| 25.
[6]	Benett, A. G., Aspherical and continuous curve contact lenses, Part Three, Optom. Today 28, 238-242, 1988
[7]	C. C. Ferguson, “Intersections of Ellipsoids and Planes of Arbitrary Orientation and Position,” Mathematical Geology, Vol. 11, No. 3, pp. 329-336. doi: 10.1007/BF01034997, 1979.
[8]	Douthwaite, W. A., Pardhan, S., Surface tilt measured with the EyeSys Videokeratoscope: influence on corneal asymmetry, Invest. Ophthalmol., 1998, Vis. Sci 39, 1727-1737.
[9]	Feltens J., Vector method to compute the Cartesian (X, Y, Z) to geodetic (φ, λ, h) transformation on a triaxial ellipsoid. J Geod 83: 129–137, 2009.
[10]	Gray, A. "The Ellipsoid" and "The Stereographic Ellipsoid." §13.2 and 13.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 301-303, 1997.
[11]	Harris W. F. Curvature of ellipsoids and other surfaces. Ophthalmic Physiol Opt 2006; 26: 497–501.
[12]	James R. C., Mathematics Dictionary, 5th Edition Number, Springer Netherlands, ISBN 978-0-412-99041-0, 1992.
[13]	Klein, P., "On the Ellipsoid and Plane Intersection Equation," Applied Mathematics, Vol. 3 No. 11, pp. 1634-1640. doi: 10.4236/am.2012.311226, 2012.
[14]	Lipschutz M, Schaum’s Outlines – Differential Geometry, Schaum’s/McGraw-Hill, 1969, ISBN 0–07–037985–8.
[15]	Ligas M., Cartesian to geodetic coordinates conversion on a triaxial ellipsoid, J. Geod., 86, 249-256. 2012.
[16]	Moritz, H., Advanced Physical Geodesy, Herbert Wichmann Verlag Karlsruhe. 1980.
[17]	Zhang Hongxin, Feng Jieqing, Preliminary Mathematics of Geometric Modeling (4), State Key Lab of CAD&CG, http://www.cad.zju.edu.cn/home/zhx/GM/003/00-sg.pdf, 2006.
[18]	URL-1 http://www.mathworks.com/matlabcentral/fileexchange/46248-converter-cartesian-coordinates-to-geodetic-coordinates.
[19]	URL-2 http://www.mathworks.com/matlabcentral/fileexchange/52958-intersection-ellipsoid-and-a-plane.

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Sebahattin Bektas. (2017). Curvature of the Ellipsoid with Cartesian Coordinates. Landscape Architecture and Regional Planning, 2(2), 61-66. https://doi.org/10.11648/j.larp.20170202.13

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ACS Style

Sebahattin Bektas. Curvature of the Ellipsoid with Cartesian Coordinates. Landsc. Archit. Reg. Plan. 2017, 2(2), 61-66. doi: 10.11648/j.larp.20170202.13

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AMA Style

Sebahattin Bektas. Curvature of the Ellipsoid with Cartesian Coordinates. Landsc Archit Reg Plan. 2017;2(2):61-66. doi: 10.11648/j.larp.20170202.13

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@article{10.11648/j.larp.20170202.13,
  author = {Sebahattin Bektas},
  title = {Curvature of the Ellipsoid with Cartesian Coordinates},
  journal = {Landscape Architecture and Regional Planning},
  volume = {2},
  number = {2},
  pages = {61-66},
  doi = {10.11648/j.larp.20170202.13},
  url = {https://doi.org/10.11648/j.larp.20170202.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.larp.20170202.13},
  abstract = {This study aims to show how to obtain the curvature of the ellipsoid depending on azimuth angle. The curvature topic is quite popular at an interdisciplinary level. It can be to the friends of geometry, geodesy, satellite orbits in space, in studying all sorts of elliptical motions (e.g., planetary motions), curvature of surfaces and concerning eye-related radio-therapy treatment, for example the anterior surface of the cornea is often represented as ellipsoidal in form. On the calculation of the curvature, there is a famous Euler formula for rotating ellipsoid that everyone knows. Let θ be the angle, in the tangent plane, measured clockwise from the direction of minimum curvature κ1. Then the normal curvature κn (θ) in direction θ is given by κn (θ) = κ1 cos2θ + κ2 sin2θ = κ1 + (κ2 - κ1) cos2θ I wonder how can a formula for a triaxial ellipsoid? So we started to work. And we finally found the formula for the triaxial ellipsoid.},
 year = {2017}
}

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TY  - JOUR
T1  - Curvature of the Ellipsoid with Cartesian Coordinates
AU  - Sebahattin Bektas
Y1  - 2017/03/04
PY  - 2017
N1  - https://doi.org/10.11648/j.larp.20170202.13
DO  - 10.11648/j.larp.20170202.13
T2  - Landscape Architecture and Regional Planning
JF  - Landscape Architecture and Regional Planning
JO  - Landscape Architecture and Regional Planning
SP  - 61
EP  - 66
PB  - Science Publishing Group
SN  - 2637-4374
UR  - https://doi.org/10.11648/j.larp.20170202.13
AB  - This study aims to show how to obtain the curvature of the ellipsoid depending on azimuth angle. The curvature topic is quite popular at an interdisciplinary level. It can be to the friends of geometry, geodesy, satellite orbits in space, in studying all sorts of elliptical motions (e.g., planetary motions), curvature of surfaces and concerning eye-related radio-therapy treatment, for example the anterior surface of the cornea is often represented as ellipsoidal in form. On the calculation of the curvature, there is a famous Euler formula for rotating ellipsoid that everyone knows. Let θ be the angle, in the tangent plane, measured clockwise from the direction of minimum curvature κ1. Then the normal curvature κn (θ) in direction θ is given by κn (θ) = κ1 cos2θ + κ2 sin2θ = κ1 + (κ2 - κ1) cos2θ I wonder how can a formula for a triaxial ellipsoid? So we started to work. And we finally found the formula for the triaxial ellipsoid.
VL  - 2
IS  - 2
ER  -

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Author Information

Sebahattin Bektas

Geomatic Engineering, Faculty of Engineering, Ondokuz Mayis University, Samsun, Turkey

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APA Style

Sebahattin Bektas. (2017). Curvature of the Ellipsoid with Cartesian Coordinates. Landscape Architecture and Regional Planning, 2(2), 61-66. https://doi.org/10.11648/j.larp.20170202.13

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ACS Style

Sebahattin Bektas. Curvature of the Ellipsoid with Cartesian Coordinates. Landsc. Archit. Reg. Plan. 2017, 2(2), 61-66. doi: 10.11648/j.larp.20170202.13

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AMA Style

Sebahattin Bektas. Curvature of the Ellipsoid with Cartesian Coordinates. Landsc Archit Reg Plan. 2017;2(2):61-66. doi: 10.11648/j.larp.20170202.13

Copy | Download

@article{10.11648/j.larp.20170202.13,
  author = {Sebahattin Bektas},
  title = {Curvature of the Ellipsoid with Cartesian Coordinates},
  journal = {Landscape Architecture and Regional Planning},
  volume = {2},
  number = {2},
  pages = {61-66},
  doi = {10.11648/j.larp.20170202.13},
  url = {https://doi.org/10.11648/j.larp.20170202.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.larp.20170202.13},
  abstract = {This study aims to show how to obtain the curvature of the ellipsoid depending on azimuth angle. The curvature topic is quite popular at an interdisciplinary level. It can be to the friends of geometry, geodesy, satellite orbits in space, in studying all sorts of elliptical motions (e.g., planetary motions), curvature of surfaces and concerning eye-related radio-therapy treatment, for example the anterior surface of the cornea is often represented as ellipsoidal in form. On the calculation of the curvature, there is a famous Euler formula for rotating ellipsoid that everyone knows. Let θ be the angle, in the tangent plane, measured clockwise from the direction of minimum curvature κ1. Then the normal curvature κn (θ) in direction θ is given by κn (θ) = κ1 cos2θ + κ2 sin2θ = κ1 + (κ2 - κ1) cos2θ I wonder how can a formula for a triaxial ellipsoid? So we started to work. And we finally found the formula for the triaxial ellipsoid.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Curvature of the Ellipsoid with Cartesian Coordinates
AU  - Sebahattin Bektas
Y1  - 2017/03/04
PY  - 2017
N1  - https://doi.org/10.11648/j.larp.20170202.13
DO  - 10.11648/j.larp.20170202.13
T2  - Landscape Architecture and Regional Planning
JF  - Landscape Architecture and Regional Planning
JO  - Landscape Architecture and Regional Planning
SP  - 61
EP  - 66
PB  - Science Publishing Group
SN  - 2637-4374
UR  - https://doi.org/10.11648/j.larp.20170202.13
AB  - This study aims to show how to obtain the curvature of the ellipsoid depending on azimuth angle. The curvature topic is quite popular at an interdisciplinary level. It can be to the friends of geometry, geodesy, satellite orbits in space, in studying all sorts of elliptical motions (e.g., planetary motions), curvature of surfaces and concerning eye-related radio-therapy treatment, for example the anterior surface of the cornea is often represented as ellipsoidal in form. On the calculation of the curvature, there is a famous Euler formula for rotating ellipsoid that everyone knows. Let θ be the angle, in the tangent plane, measured clockwise from the direction of minimum curvature κ1. Then the normal curvature κn (θ) in direction θ is given by κn (θ) = κ1 cos2θ + κ2 sin2θ = κ1 + (κ2 - κ1) cos2θ I wonder how can a formula for a triaxial ellipsoid? So we started to work. And we finally found the formula for the triaxial ellipsoid.
VL  - 2
IS  - 2
ER  -

Copy | Download