W--line congruences

Abstract

By utilizing the Darboux frames, along with a regular surface whose parametric curves are lines of curvature, we analyzed the normal line congruence which preserves the asymptotic curves between its focal surfaces. This allows deriving systems of partial differential equations through which the problem of determining the director surface and the corresponding normal line congruence could be solved. Moreover, a necessary and sufficient condition that the focal surfaces of the normal line congruence are degenerates into curves is derived. As a result, the middle focal surface of the normal line congruence is presented as a new surface interrogation tool.

Keywords

• Lines of curvature,W-Line congruence,Area preserving representation

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