Pacific Graphics 2011 Bing-Yu Chen, Jan Kautz, Tong-Yee Lee, and Ming C. Lin (Guest Editors) Volume 30 (2011), Number 7 Optimized Topological Surgery for Unfolding 3D Meshes Shigeo Takahashi 1 , Hsiang-Yun Wu 1 , Seow Hui Saw 1 , Chun-Cheng Lin 2 , and Hsu-Chun Yen 3 1 The University of Tokyo, Japan 2 National Chiao Tung University, Taiwan 3 National Taiwan University, Taiwan Abstract Constructing a 3D papercraft model from its unfolding has been fun for both children and adults since we can reproduce virtual 3D models in the real world. However, facilitating the papercraft construction process is still a challenging problem, especially when the shape of the input model is complex in the sense that it has large variation in its surface curvature. This paper presents a new heuristic approach to unfolding 3D triangular meshes without any shape distortions, so that we can construct the 3D papercraft models through simple atomic operations for gluing boundary edges around the 2D unfoldings. Our approach is inspired by the concept of topological surgery, where the appearance of boundary edges of the unfolded closed surface can be encoded using a symbolic representation. To fully simplify the papercraft construction process, we developed a genetic-based algorithm for unfolding the 3D mesh into a single connected patch in general, while optimizing the usage of the paper sheet and balance in the shape of that patch. Several examples together with user studies are included to demonstrate that the proposed approach works well for a broad range of 3D triangular meshes. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geome- try and Object Modeling—Geometric algorithms, languages, and systems - mesh unfolding, topological surgery, genetic algorithms, papercraft models 1. Introduction Unfolding 3D meshes into 2D papercraft models allows us to retrieve the corresponding 3D physical shapes from the 2D display to the real world. This technique would be very useful when we prepare miniatures of 3D scenes, such as ar- chitectural designs and urban plannings using the papercraft models. Furthermore, regardless of the recent development of rapid prototyping, constructing the 3D physical models from 2D papercraft patterns is by itself an interesting enter- tainment, especially for children and families to share pleas- ant experiences. A variety of methods have been developed recently for this purpose both in the fields of computational geometry and computer graphics. Although existing techniques are effective, they usually decompose an input 3D model into a relatively large num- ber of unfolded patches, including small pieces as shown in Figure 1(a). This is a serious problem in practice because we have to seek the correspondences between the bound- ary edges of different patches when merging them, or even worse, we have to fit a tiny piece, having a few faces only, to the remaining part of a 3D papercraft model tightly enough. Actually, facilitating simple search for the boundary edge matching is crucial for accelerating the construction of the corresponding papercraft model. This paper presents a new heuristic approach for fully op- timizing the 2D unfolding of an input 3D mesh. The key ideas of our approach are inspired by the concept of topolog- ical surgery, which allows us to encode the edge sequence on the boundary of the unfolding using a symbolic repre- sentation. Furthermore, in order to ease the papercraft con- struction process, we have developed a genetic-based algo- rithm for unfolding the 3D mesh into one single patch. This formulation allows us to construct the 3D papercraft model only through simple atomic operations for merging bound- ary edges of the 2D unfolding, where we can always find a pair of duplicated edges that are next to each other. Our approach is distortion-free in the sense that we need not stretch and shrink unfolded patterns to construct the papercraft models, unlike most conventional approaches. Actually, unfolding a polyhedron into a single unfolded patch without any deformation is a well-known open prob- lem [She75, DO05] and has intensively been studied so far c  2011 The Author(s) Computer Graphics Forum c  2011 The Eurographics Association and Blackwell Publish- ing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. DOI: 10.1111/j.1467-8659.2011.02053.x