Bertrand framed surfaces in the Euclidean 3-space and its applications

2025/05/07 by Nozomi Nakatsuyama, Masatomo Takahashi, Nakatsuyama, Nozomi +1 · 1 citation
Engineering · Mathematics · #53A05 #57R45 #58K05 #Advanced Numerical Analysis Techniques #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2505.04124

openalex publication_date 2025/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A framed surface is a smooth surface in the Euclidean space with a moving frame. By using the moving frame, we can define Bertrand framed surfaces as the same idea as Bertrand framed curves. Then we find the caustics and involutes as Bertrand framed surfaces. As applications, we can directly define the caustics and involutes of framed surfaces, and give conditions that the caustics and involutes are inverse operations of framed surfaces like as those of Legendre curves. Moreover, a framed surface is one of the Bertrand framed surfaces if and only if another caustic of the involute exists, under conditions. Furthermore, we find a new such operation, the so-called tangential direction framed surfaces.

Citations

• Bertrand types of regular curves and Bertrand framed curves in the Euclidean 3-space
• Focal surfaces of fronts associated to unbounded principal curvatures

Cited by

• Envelopes created by sphere families in Euclidean 3-space

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