An Overview of Developable Surfaces in Geometric Modeling

Background: A developable surface is a special ruled surface with vanishing Gaussian curvature. The study of developable surfaces is of interest in both academia and industry. The application of developable surfaces ranges from ship hulls, architecture to origami, clothes, etc., as they are suitable for the modeling of surfaces with materials that are not amenable to stretch like leather, paper, fiber, and sheet metal. Objective: We survey techniques and patents of developable surfaces in the field of geometric modeling. The theory, algorithms, and applications are discussed to provide a comprehensive summary for modeling developable surfaces. Methods: Prior methods that model smooth and discrete developable surfaces in diverse disciplines are collected and reviewed. In particular, our review focuses on C², C¹ and C⁰ developable surfaces, which are driven by the problems and challenges in the industry. Results: Many papers and patents of developable surface modeling are classified in this review paper. It is found that remarkable developments and improvements have been achieved in both analytical computations and practical applications. Conclusion: Global piecewise-developable surfaces, exploration of shape space of developable surfaces, joint optimization of geometry and physics, and other fundamental problems should be further studied.