Surface discretisation with rectifying strips on Geodesics

The use of geodesic curves of surfaces has enormous potential in architecture due to their multiple properties and easy geometric control using digital graphic tools. Among their numerous properties, the geodesic curves of a surface are the paths along which straight strips can be placed tangentially to the surface. On this basis, a graphical method is proposed to discretize surfaces using straight strips, which optimizes material consumption since rectangular straight strips take advantage of 100% of the material in the cutting process. The contribution of the article consists of presenting the geometric constraints that characterize this type of panelling from the idea of “rectifying surface”, considering the material inextensible. Experimental prototypes that have been part of the research are also described and the final theoretical results are presented

Developable Quad Meshes

There are different ways to capture the property of a surface being developable, i.e., it can be mapped to a planar domain without stretching or tearing. Contributions range from special parametrizations to discrete-isometric mappings. So far, a local criterion expressing the developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is based on a property well-known from differential geometry, namely a rank-deficient second fundamental form. This criterion is expressed in terms of the canonical checkerboard patterns inscribed in a quad mesh which already was successful in describing discrete-isometric mappings. In combination with standard global optimization procedures, we are able to perform developable lofting, approximation, and design. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves. They are thus easily embedded into a design workflow involving standard operations like re-meshing, trimming, and merging operations

A Survey of Developable Surfaces: From Shape Modeling to Manufacturing

Developable surfaces are commonly observed in various applications such as architecture, product design, manufacturing, and mechanical materials, as well as in the development of tangible interaction and deformable robots, with the characteristics of easy-to-product, low-cost, transport-friendly, and deformable. Transforming shapes into developable surfaces is a complex and comprehensive task, which forms a variety of methods of segmentation, unfolding, and manufacturing for shapes with different geometry and topology, resulting in the complexity of developable surfaces. In this paper, we reviewed relevant methods and techniques for the study of developable surfaces, characterize them with our proposed pipeline, and categorize them based on digital modeling, physical modeling, interaction, and application. Through the analysis to the relevant literature, we also discussed some of the research challenges and future research opportunities.Comment: 20 pages, 24 figures, Author submitted manuscrip