General Mathematics

    

From Piecewise-Linear to Piecewise-Circular: A Geometric Extension of Integration Theory via Osculating Sectors

Authors: Giuseppe D'Ambrosio

This paper introduces the D’Ambrosio Integral, a geometrically oriented generalization of the Riemann integral that incorporates the curvature of the integrable curve. Unlike classical schemes based on linear approximation, the proposed method employs circular arcs determined by consecutive triplets of points, allowing curvature information to be directly embedded into the integration process. It is shown that the integral converges, is geometrically invariant, and coincides with the Riemann integral for sufficiently smooth functions. The formulation naturally extends to Riemannian manifolds through the use of the exponential map. The proposed approach provides higher geometric accuracy in regions of pronounced curvature and is applicable to the analysis of curves, vector fields, and dynamical systems.

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[v1] 2026-05-20 22:18:45

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