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.
Highlights
In Euclidean 3-space, a two-parameter set of lines is called a line congruence
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
As a result the middle focal surface of the normal line congruence is presented as a new surface interrogation tool
Summary
In Euclidean 3-space, a two-parameter set of lines is called a line congruence. For instance, the normal vector ...eld of a surface constitute such a line congruence but this is not the general situation. A necessary and su¢ cient condition that the focal surfaces of the normal line congruence are degenerates into curves has been derived. In general it consists of two sheets, conjugated to the two families of lines of curvature and called focal surfaces of M .
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