Developable Ruled Surfaces Generated by the Curvature Axis of a Curve
Ferhat Taş, Rushan Ziatdinov
TL;DR
This paper shows that a ruled surface generated by moving the curvature axis of a space curve is necessarily developable, establishing a direct curve–surface correspondence via $\mathbf{R}(s,\lambda)=\boldsymbol\alpha(s)+\frac{1}{\kappa(s)}\mathbf{n}(s)+\lambda\mathbf{b}(s)$. It analyzes conditions under which a given developable surface arises from generalized cylinder, conical, or tangent forms, deriving explicit constraints such as $\mathbf{R}(s,\lambda)$ forms, $\alpha''(s)$ relations, and when $\kappa$ and $\tau$ take specific values (e.g., tangent developables require $\kappa$ constant and $\tau=\kappa$). The work further explores singularities of developables—cuspidal-edge, swallowtail, and cuspidal cross-cap—and demonstrates how singular space curves induce singular developable surfaces, with visualisation guidance using environment maps. Overall, the study advances differential-geometry-based design rules for CAD/CAM by linking curvature-axis motion to practical, developable surface generation, highlighting potential applications in architecture and manufacturing and proposing future integration with advanced curve families like log-aesthetic curves and superspirals.
Abstract
Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of much discussion in mathematics and engineering journals. In geometric modelling, ideas are successful if they are not too complex for engineers and practitioners to understand and not too difficult to implement, because these specialists put mathematical theories into practice by implementing them in CAD/CAM systems. Some of these popular systems such as AutoCAD, Solidworks, CATIA, Rhinoceros 3D, and others are based on simple polynomial or rational splines and many other beautiful mathematical theories that have not yet been implemented due to their complexity. Based on this philosophy, in the present work, we investigate a simple way to generate ruled surfaces whose generators are the curvature axes of curves. We show that this type of ruled surface is a developable surface and that there is at least one curve whose curvature axis is a line on the given developable surface. In addition, we discuss the classifications of developable surfaces corresponding to space curves with singularities, while these curves and surfaces are most often avoided in practical design. Our research also contributes to the understanding of the singularities of developable surfaces and, in their visualisation, proposes the use of environmental maps with a circular pattern that creates flower-like structures around the singularities.