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Axioms
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14 December 2025

On the Focal Geometry of Translation Surfaces: Flatness, Minimality, and Classification Results

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1
Department of Mechanics, Kocaeli University, 41650 Kocaeli, Türkiye
2
Department of Mathematics, Kocaeli University, 41380 Kocaeli, Türkiye
3
Department of Mathematics, İzmir Democracy University, 35140 İzmir, Türkiye
4
Department of Mathematics, Sakarya University, 54187 Sakarya, Türkiye
Axioms2025, 14(12), 919;https://doi.org/10.3390/axioms14120919 
(registering DOI)
This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition

Abstract

In this study, we investigate the focal surfaces associated with translation surfaces in Euclidean 3-space from the viewpoint of differential geometry. We begin by defining the translation surface generated by two planar curves and derive the corresponding focal surfaces using the framework of the Frenet frame. Analytical conditions are obtained under which the focal surfaces exhibit minimality or flatness. Several theorems are proven to classify the focal images, supported by illustrative examples. The results provide insights into the curvature structure of translation surfaces and contribute to the broader understanding of their geometric behavior.

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