HOLDITCH THEOREM FOR THE CLOSED SPACE CURVES IN LORENTZIAN 3-SPACE

doi:10.1016/S0252-9602(11)60218-X

Abstract

Abstract:

In this article, we give the area formula of the closed projection curve of a closed space curve in Lorentzian 3-space L³. For the 1-parameter closed Lorentzian space motion in L³, we obtain a Holditch Theorem taking into account the Lorentzian matrix multiplication for the closed space curves by using their othogonal projections onto the Euclidean plane in the fixed Lorentzian space. Moreover, we generalize this Holditch Theorem for noncollinear three fixed points of the moving Lorentzian space and any other fixed point on the plane which is determined by these three fixed points.

Key words: Lorentzian matrix multiplication, Lorentzian motion, Holditch Theorem, orthogonal projection area

CLC Number:

53A17

Handan Yildirim, Salim Y¨uce, Nuri Kuruoglu. HOLDITCH THEOREM FOR THE CLOSED SPACE CURVES IN LORENTZIAN 3-SPACE[J].Acta mathematica scientia,Series B, 2011, 31(1): 172-180.

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References

[1] Holditch H. Geometrical theorem. Q J Pure Appl Math, 1858, 2: 38

[2] Broman A. Holditch's theorem is somewhat deeper than Holditch thought in 1858. Normat, 1979, 3: 89--100

[3] Broman A. Holditch's theorem. Math Mag, 1981, 54(3): 99--108

[4] Tutar A, Kuruo\u{g}lu N. The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics. Mech Mach
Theory, 1999, 34: 1--6

[5] M\"{u}ller H R. Erweiterung des Satzes von Holditch f\"{u}r geschlossene Raumkurven. Abh Braunschw Wiss Ges, 1980, 31: 129-135

[6] M\"{u}ller H R. Ein Holditch-Satz f\"{u}r Fl\"{a}chenst\"{u}cke im R³. Abh Braunschw Wiss Ges 1987, 39: 37-42

[7] Pottmann H. Holditch-Sicheln. Arch Math, 1985, 44: 373--378

[8] Pottmann H. Zum Satz von Holditch in der euklidischen Ebene. Elem Math, 1986, 41: 1--6

[9] Hoschek J. Eine Verallgemeinerung des Satzes von Holditch. Monatsh Math, 1975, 80: 93--99

[10] Hering L. S\"{a}tze vom Holditch-Typ f\"{u}r ebene Kurven. Elem Math, 1983, 38: 39--49 (In German)

[11] Kuruo\u{g}lu N, Y\"{u}ce S. The generalized Holditch theorem for the homothetic motions on the planar kinematics. Czechoslovak Math J,
2004, 54(129): 337--340

[12] Hentschke S. Erweiterungen des Satzes von Hoditch. Sitzungsber Österr Akad Wiss, 1975, 184: 451--458 (In German)

[13] Blaschke W, M\"{u}ller H R. Ebene Kinematik. German: Verlag Oldenbourg, M\"{u}nchen, 1956

[14] O' Neill B. Semi-Riemannian Geometry. New York: Academic Press, 1983

[15] G\"{u}ndo\u{g}an H, Ke\c{c}ilio\u{g}lu O. Lorentzian matrix multiplication and the motions on Lorentzian plane. Glasnik Matematicki,
2006, 41(61): 329--334

[16] Greub W H. Linear Algebra. 3rd ed. New York: Springer-Verlag and Academic Press, 1967

[17] Dzan J J. Trigonometric Laws on Lorentzian Sphere S₁². Journal of Geometry, 1985, 24: 6--13