Ortho-arc Curves: Generative Design of Developable Bands and Polygonal Tubes

The design of developable surfaces from closed directrices with prescribed symmetry and non-trivial topology is usually treated through iterative optimization. We introduce ortho-arc curves (OACs), C 1 piecewise-circular curves generated from polygonal paths on a support surface, as part of a forward analytic approach distinct from inverse shape approximation. For closed OACs, we show that rotation-minimizing-frame closure is governed by quantized signed accumulated torsion, and use this criterion to generate one-parameter families of patchwise exactly developable bands. The resulting bands are assembled from planar, cylindrical, and conical patches joined with C 1 continuity globally. With compatible polygonal cross-sections, we further construct polygonal tubes with developable faces. Through polyhedral precursors, we control the geometry and topology of the OAC directrices by construction, and from spherical duality and indicatrix deformations we obtain continuous families with fixed orientability type. Beyond the core construction, we prove that the ortho-arc condition and rotation-minimizing frame holonomy are preserved under Möbius transformations in R 3 , with chirality reversal under orientation-reversing maps. Finally, we illustrate knotted and linked topologies and outline exploratory extensions to reciprocal-frame-like cylinder scaffolds, piecewise-helical directrices, and higher-regularity directrices retaining the properties of precursor-driven topology and frame-holonomy control. These results provide a generative route to developable band and polygonal tube design for fabrication-oriented workflows.