On the Line Congruences

Ferhat Taş; Osman Gürsoy

doi:10.36890/iejg.545123

Research Article

Year 2018, Volume: 11 Issue: 2, 47 - 53, 30.11.2018

Ferhat Taş Osman Gürsoy

https://doi.org/10.36890/iejg.545123

Cited By: 5

Abstract

References

[1] Abdel-Baky, R.A., Al-Bokhary, A.J., A new approach for describing instantaneous line congruence, Archivum Mathematicum, (2008), pp. 223-236.
[2] Ball, R.S., A Treatise on the Theory of Screws, Cambridge University Press, 1900 (reprinted 1999).
[3] Blaschke, W., Vorlesungen über differential geometrie I, Springer-Verlag, 1945, Berlin.
[4] Bottema, O., Roth, B., Theoretical Kinematics, Dover Publications, 1990.
[5] Clifford, W.K., Preliminary sketch of bi-quaternions, Proceedings of the Royal Society of London, (1873), pp. 381-395.
[6] Dimentberg, F.M., The Screw Calculus and Its Applications in Mechanics, Foreign Technology Division translation, FTD-HT-1632-67, 1965.
[7] Eisenhart, L.P., A Treatise in Differential Geometry of Curves and Surfaces, New York, Ginn Camp., 1969.
[8] Guggenheimer, H.W., Differential geometry, Dover Publications, Inc. New York, 1977.
[9] Gürsoy, O. J Geom (1990) 39: 80. https://doi.org/10.1007/BF01222141.
[10] Gürsoy, O., Some results on closed ruled surfaces and closed space curves, Mech. Mach. Theory, (1992), pp.323-330.
[11] Hlavaty, V., Differential line geometry, Groningen, P.Noordhoff Ltd. X, 1953.
[12] Hoschek, J., Liniengeometrie, B.I. Hochschultaschenbuch, Mannheim, 1971.
[13] Hunt, K.H., Kinematic Geometry of Mechanisms, Oxford Press, 1978.
[14] Jüttler, B., Rittenschober, K., Using line congruences for parametrizing special algebraic surfaces, The Mathematics of Surfaces X, Lecture Notes in Computer Science, vol.2768, Springer-Berlin, 2003, pp. 223-243.
[15] Köse, Ö., A method of the determination of a developable ruled surface, Mech. Mach. Theory, (1999), pp. 1187-1193.
[16] Larochelle, P.M., Vance, J.M., Kihonge, J.N., Interactive visualization of line congruences for spatial mechanism design, Journal of Computing and Information Science in Engineering, (2002), pp. 208-215.
[17] Odehnal, B., On rational isotropic congruences of lines, Journal of Geometry, (2004), pp. 126-138.
[18] Plücker, J., On a new geometry of space, Proceedings of the Royal Society of London, (1865), pp. 53-58.
[19] Pottmann, H., Wallner, J., Computational line geometry, Mathematics and Visualization, Springer-Verlag Berlin Heidelberg, 2001.
[20] Selig, J.M., Geometrical Methods in Robotics, Springer, New York, 1994.
[21] Shepherd, M.D., Line congruences as surfaces in the space of lines, Differential Geometry and its Applications, (1999), pp. 1-26.
[22] Study, E., Geometry der dynamen, Leipzip, 1903.
[23] Veldkamp, G.R., On the use of dual numbers vectors and matrices in instantaneous spatial kinematics, Mech. Mach. Theory, (1976), pp. 141-156.

On the Line Congruences

Year 2018, Volume: 11 Issue: 2, 47 - 53, 30.11.2018

Ferhat Taş Osman Gürsoy

https://doi.org/10.36890/iejg.545123

Cited By: 5

Abstract

The purpose of this paper is to find the quantities and surfaces of a line congruence via

examining it in the dual space and to represent the results more appropriately for computational

approximations. For this, we take mainly two-dual parameter motion on the dual unit sphere (DUS)

so, we get a line congruence corresponding this motion by a new method. Thus, the equations of

the developable surfaces, the principal surfaces, the focal surfaces and the center surface of the line

congruence are found by coordinate functions. The results are illustrated by examples.

Keywords

Differential line geometry, Ruled surface, Developable ruled surface

References

[1] Abdel-Baky, R.A., Al-Bokhary, A.J., A new approach for describing instantaneous line congruence, Archivum Mathematicum, (2008), pp. 223-236.
[2] Ball, R.S., A Treatise on the Theory of Screws, Cambridge University Press, 1900 (reprinted 1999).
[3] Blaschke, W., Vorlesungen über differential geometrie I, Springer-Verlag, 1945, Berlin.
[4] Bottema, O., Roth, B., Theoretical Kinematics, Dover Publications, 1990.
[5] Clifford, W.K., Preliminary sketch of bi-quaternions, Proceedings of the Royal Society of London, (1873), pp. 381-395.
[6] Dimentberg, F.M., The Screw Calculus and Its Applications in Mechanics, Foreign Technology Division translation, FTD-HT-1632-67, 1965.
[7] Eisenhart, L.P., A Treatise in Differential Geometry of Curves and Surfaces, New York, Ginn Camp., 1969.
[8] Guggenheimer, H.W., Differential geometry, Dover Publications, Inc. New York, 1977.
[9] Gürsoy, O. J Geom (1990) 39: 80. https://doi.org/10.1007/BF01222141.
[10] Gürsoy, O., Some results on closed ruled surfaces and closed space curves, Mech. Mach. Theory, (1992), pp.323-330.
[11] Hlavaty, V., Differential line geometry, Groningen, P.Noordhoff Ltd. X, 1953.
[12] Hoschek, J., Liniengeometrie, B.I. Hochschultaschenbuch, Mannheim, 1971.
[13] Hunt, K.H., Kinematic Geometry of Mechanisms, Oxford Press, 1978.
[14] Jüttler, B., Rittenschober, K., Using line congruences for parametrizing special algebraic surfaces, The Mathematics of Surfaces X, Lecture Notes in Computer Science, vol.2768, Springer-Berlin, 2003, pp. 223-243.
[15] Köse, Ö., A method of the determination of a developable ruled surface, Mech. Mach. Theory, (1999), pp. 1187-1193.
[16] Larochelle, P.M., Vance, J.M., Kihonge, J.N., Interactive visualization of line congruences for spatial mechanism design, Journal of Computing and Information Science in Engineering, (2002), pp. 208-215.
[17] Odehnal, B., On rational isotropic congruences of lines, Journal of Geometry, (2004), pp. 126-138.
[18] Plücker, J., On a new geometry of space, Proceedings of the Royal Society of London, (1865), pp. 53-58.
[19] Pottmann, H., Wallner, J., Computational line geometry, Mathematics and Visualization, Springer-Verlag Berlin Heidelberg, 2001.
[20] Selig, J.M., Geometrical Methods in Robotics, Springer, New York, 1994.
[21] Shepherd, M.D., Line congruences as surfaces in the space of lines, Differential Geometry and its Applications, (1999), pp. 1-26.
[22] Study, E., Geometry der dynamen, Leipzip, 1903.
[23] Veldkamp, G.R., On the use of dual numbers vectors and matrices in instantaneous spatial kinematics, Mech. Mach. Theory, (1976), pp. 141-156.

There are 23 citations in total.

Details

Primary Language	English
Journal Section	Research Article
Authors	Ferhat Taş Osman Gürsoy This is me
Publication Date	November 30, 2018
Published in Issue	Year 2018 Volume: 11 Issue: 2

Cite

APA	Taş, F., & Gürsoy, O. (2018). On the Line Congruences. International Electronic Journal of Geometry, 11(2), 47-53. https://doi.org/10.36890/iejg.545123
AMA	Taş F, Gürsoy O. On the Line Congruences. Int. Electron. J. Geom. November 2018;11(2):47-53. doi:10.36890/iejg.545123
Chicago	Taş, Ferhat, and Osman Gürsoy. “On the Line Congruences”. International Electronic Journal of Geometry 11, no. 2 (November 2018): 47-53. https://doi.org/10.36890/iejg.545123.
EndNote	Taş F, Gürsoy O (November 1, 2018) On the Line Congruences. International Electronic Journal of Geometry 11 2 47–53.
IEEE	F. Taş and O. Gürsoy, “On the Line Congruences”, Int. Electron. J. Geom., vol. 11, no. 2, pp. 47–53, 2018, doi: 10.36890/iejg.545123.
ISNAD	Taş, Ferhat - Gürsoy, Osman. “On the Line Congruences”. International Electronic Journal of Geometry 11/2 (November 2018), 47-53. https://doi.org/10.36890/iejg.545123.
JAMA	Taş F, Gürsoy O. On the Line Congruences. Int. Electron. J. Geom. 2018;11:47–53.
MLA	Taş, Ferhat and Osman Gürsoy. “On the Line Congruences”. International Electronic Journal of Geometry, vol. 11, no. 2, 2018, pp. 47-53, doi:10.36890/iejg.545123.
Vancouver	Taş F, Gürsoy O. On the Line Congruences. Int. Electron. J. Geom. 2018;11(2):47-53.