Dual Ribbon Frames and the Geometry of Ribbon Surfaces[i]
In this paper, we introduce the dual ribbon frame, a differentiable moving frame that avoids the singularities inherent in the Frenet–Serret system, and develop its dual representation for ribbon surfaces. By transferring the ribbon vectors of a unit-speed curve on the dual unit sphere to the origin, we construct dual spherical representation curves, which correspond to ruled surfaces in Euclidean space via the Study mapping. We then characterize the Gaussian curvatures of these ruled surfaces and determine conditions under which they are developable. Our results extend the existing dual geometric framework from Frenet and Bishop frames to ribbon frames, providing a new perspective for the study of ruled and developable surfaces. The dual ribbon approach not only enriches the theory of differential geometry but also has potential applications in geometric modeling, computer-aided design, and the analysis of ribbon-like structures in physics and biology.
Keywords: Frenet frame, Ribbon frame, dual space, dual numbers.
[i] Dual Representation and Geometric Analysis of Ribbon Surfaces
