Some Aspects of the Geometry of Surfaces with a System of Flat Coordinate Lines
V.N. Ivanov1, GIL-OULBE Mathieu2
1Dr. V.N. Ivanov, Dsc. civil engineering, professor.
2Dr. GIL-OULBE Mathieu, PHD civil engineering, associate professor.
Manuscript received on March 01, 2014. | Revised Manuscript received on March 03, 2014. | Manuscript published on March 05, 2014. | PP: 77-82 | Volume-4 Issue-1, March 2014. | Retrieval Number: F2008013614/2014©BEIESP
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Abstract: The geometry of surfaces with system of flat coordinate lines is investigated. The general vector equation of surfaces obtained. On the basis of which formulas of the main quadratic forms are obtained. The condition at which performance the system of flat coordinate lines will be system of the main lines of curvature is obtained. The subclass of normal surfaces – surfaces with system of flat coordinate lines in the normal planes of directrix is studied. As an example on the basis of the general formulas the equation of Monge’s ruled surfaces is obtained. Drawings of surfaces of Monge with various flat and spatial directrices and various flat generatrices are given.
Keywords: Surfaces with system of flat coordinate lines, the vector equation, coefficients of quadratic forms, lines of main curvatures, normal surfaces, Monge’s ruled surfaces.