A curvature-dependent adapted frame generated by Frenet and Bishop normals

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Abstract

We studied a curvature-dependent adapted frame, denoted for brevity by the adaptive Bishop–Frenet (ABF) frame, on Frenet-regular intervals where the curvature is strictly positive. The name refers only to the fact that the defining vector is generated explicitly from the Frenet and Bishop normals; it does not denote an endpoint-to-endpoint interpolation. Exact normalization and a uniform nondegeneracy margin ensure that the resulting adapted frame is orthonormal on this domain. The construction neither reaches the Bishop frame as an endpoint nor extends through inflection points. The contribution is local and structural: we establish the exact normalization and its uniform lower bound, derive the two curvature coefficients and the normal-plane rotation coefficient, record the associated adapted-frame structural system, and compute the exact squared-derivative energy density. The reconstruction result included in the paper concerns general adapted orthonormal frames and is not an ABF realization theorem. No universal full-energy ordering or numerical validation is claimed. Analytical examples illustrate the resulting formulas and clarify the restricted domain of validity.

Keywords

adapted orthonormal frames / Frenet–Serret frame / Bishop frame / curvature-dependent frames / normal-plane rotation / frame energy

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