Focal surfaces of developable ruled surfaces and their invariants
DOI:
https://doi.org/10.55630/mem.2026.55.049-059Keywords:
Frenet curves, Focal curves, Ruled surfaces, Developed surfaces, Focal surfaces, Gauss curvature, Mean curvatureAbstract
In this work the focal surfaces of developable ruled surfaces generated by a unit speed Frenet curve and its focal curve in the Euclidean three-dimensional space are investigated. For each curve, we construct the associated tangential surface and derive a parametrization of its focal surface. Working with respect to the natural parameter of the base curve, we obtain relations between the coefficients of the two fundamental forms of the original developable surfaces and those of their focal surfaces. The focal surfaces obtained in both cases are themselves developable. Formulas relating to their mean curvatures are established.
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