A Survey of Developable Surfaces: From Shape Modeling to Manufacturing

Abstract: Developable surfaces are commonly observed in various applications such as architecture, product design, manufacturing, mechanical materials, and data physicalization 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.

• The paper introduces comprehensive techniques for segmenting and flattening 3D shapes with minimal distortion to generate usable developable surfaces.
• It details manufacturing processes, including subtractive and additive methods, highlighting trade-offs in speed, cost, and accuracy.
• The study emphasizes interactive assistance in design and assembly, outlining future research directions using AI for improved physical realization.

A Survey of Developable Surfaces: From Shape Modeling to Manufacturing

Developable surfaces have become an essential concept in various fields such as architecture, manufacturing, and product design. These surfaces offer advantages such as ease of production, low cost, and transportability. The paper "A Survey of Developable Surfaces: From Shape Modeling to Manufacturing" provides a comprehensive overview of the current methodologies and challenges associated with developable surfaces.

Introduction

Developable surfaces are characterized by zero Gaussian curvature and can be easily flattened into a plane without distortion. This property makes them especially useful in applications that require materials to be transported and assembled efficiently. The complexity of transforming 3D shapes into developable surfaces involves several processes, including shape segmentation, surface flattening, manufacturing, and assembly.

Figure 1: Pipeline of developable surfaces from digital modeling to physical modeling. Shape segmentation, surface flattening, digital and physical modeling, assembly, and interactive assistance.

Digital Modeling

Shape Segmentation

Shape segmentation is a critical step in the digital modeling of developable surfaces. It involves dividing complex shapes into segments that can be flattened with minimal distortion. The paper categorizes shape segmentation methods into several types: subdivision, geodesic, spanning tree, Gauss map, vector field, curvature, origami, and custom. Each method offers unique characteristics optimized for different shape and manufacturing requirements.

Figure 2: 3D shapes segmentation by subdivision: triangular meshes clustering.

Surface Flattening

Surface flattening seeks to minimize errors when transforming 3D surfaces into 2D planes. The paper discusses various optimization techniques to maintain mesh properties such as edge lengths, positions, angles, and areas. By using techniques like least squares, gradient descent, and Newton's method, researchers can achieve surface flattening with reduced distortion.

Physical Modeling

Manufacturing

The manufacturing process for developable surfaces can be categorized into subtractive and additive methods. Subtractive methods like cutting and milling are effective for flat materials, while additive methods like 3D printing and casting allow for more complex shapes. Each method offers specific trade-offs in terms of speed, cost, accuracy, and material versatility.

Figure 3: Selected samples of physical modeling: Cutting, Milling, Casting, 3D-printing, Knitting.

Assembly

Assembly processes such as folding, joint, and woven methods enable the construction of complex shapes from planar segments. This approach helps reduce fabrication challenges, allowing the rapid prototyping of intricate designs using simple manufacturing techniques.

Interactive Assistance

Interactive assistance is critical in refining the design process for developable surfaces, enabling precise control over segmentation and flattening through user interaction. Techniques such as cutting, unfolding, and parameterizing help users adjust models for better manufacturing outcomes.

Applications

Developable surfaces are widely applied across various domains. In architecture, they allow the construction of complex freeform structures with discrete panels. In product design, developable surfaces offer new aesthetic and functional possibilities. Additionally, they're explored in arts, garment design, mechanical materials, and data physicalization, showcasing the versatility and potential of developable surfaces in innovative applications.

Figure 4: Selected samples of design applications: shell construction, wearable device, origami, peeling art, origami robots, anatomical physical visualization, jewelry design of data physicalization.

Challenges and Future Work

Despite the numerous advantages and applications of developable surfaces, there are challenges concerning segmentation quality, complexity of operations, and digitization to physicalization. Future research directions include exploring new segmentation methods, improving interaction techniques, and bridging the gap between digital models and physical fabrication using AI and machine learning.

Figure 5: Expanded research based on developable surfaces: path printing by robotics for quick construction of shells.

Conclusion

The paper provides a detailed examination of developable surfaces, their methods, challenges, and applications. Developable surfaces offer significant potential across different sectors by enhancing manufacturing processes and facilitating innovative designs.

This survey represents an important step toward understanding and harnessing the capabilities of developable surfaces in practical applications, providing insights and future directions for researchers and practitioners in the field.