Inextensible Curve Flows in Dual Space D3: A Quasi-Frame Approach

This paper develops a comprehensive quasi-frame theory for inextensible curve flows in the dual space D3. While the Frenet frame has been the traditional tool for studying curve evolution, it encounters singularities when curvature vanishes. We introduce a quasi-frame (q-frame) formulation that overcomes these limitations and provides a robust framework for analyzing inextensible flows, flows that preserve arc length throughout evolution. We derive the fundamental evolution equations for quasi-curvatures under inextensible flow conditions and establish necessary and sufficient conditions for inextensibility in dual space. In contrast to the Euclidean, Minkowski, and Galilean settings, the quasi-curvatures in dual space are dual-valued, so they simultaneously encode the direction of the oriented line and the evolution of its moment vector. Our approach extends inextensible flows using the Frenet frame by providing a more general framework that remains well-defined even at singular points. We demonstrate the advantages of our method through detailed worked examples, including complete computations of time-evolving curves with explicit verification of arc-length preservation. These results contribute to the theoretical foundations of curve dynamics in dual spaces.