A Kind of Rectilinear Congruences in the Minkowski 3-Space |
Received:May 28, 2007 Revised:March 08, 2008 |
Key Words:
the rectilinear congruence the focal surfaces the Minkowski space the space-like surface maximal surfaces.
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Fund Project:Specialized Research Fund for the Doctoral Program of Higher Education (No.20050141011). |
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Abstract: |
We consider the rectilinear congruence $T$ generated by the tangents to a one parameter family of geodesics on a space-like surface $S_1$ in the Minkowski $3$-space $E_1^3$, having $S_1$ as one of its focal surfaces. We prove that the two families of torsal surfaces of $T$ touch the second focal surface $S_2$ along the net of orthogonal parametric curves if and only if $S_1$ is developable. We also obtain the necessary and sufficient condition for the correspondence between the points of $S_1$ and $S_2$ at the same rays preserving the net of asymptotic curves. At last, we investigate the orthogonal surface $S$ of $T$. We proved that the correspondence between $S_1$ and $S_2$ preserves the net of asymptotic curves if $S$ is maximal in $E_1^3$. |
Citation: |
DOI:10.3770/j.issn:1000-341X.2008.04.022 |
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