Abstract
We present a simple yet effective method for constructing 3D self-supporting surfaces with planar quadrilateral (PQ) elements. Starting with a triangular discretization of a self-supporting surface, we first compute the principal curvatures and directions of each triangular face using a new discrete differential geometry approach, yielding more accurate results than existing methods. Then, we smooth the principal direction field to reduce the number of singularities. Next, we partition all faces into two groups in terms of principal curvature difference. For each face with small curvature difference, we compute a stretch matrix that turns the principal directions into a pair of conjugate directions. For the remaining triangular faces, we simply keep their smoothed principal directions. Finally, applying a mixed-integer programming solver to the mixed principal and conjugate direction field, we obtain a planar quadrilateral mesh. Experimental results show that our method is computationally efficient and can yield high-quality PQ meshes that well approximate the geometry of the input surfaces and maintain their self-supporting properties.
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Acknowledgements
We would like to thank the anonymous reviewers for their constructive comments. This work was partially supported by National Natural Science Foundation of China (62172257, 61772312, 61772016, 61802228), Singapore Ministry of Education (T2EP20220-0014), and the RIE2020 Industry Alignment Fund–Industry Collaboration Projects (IAF–ICP) Funding Initiative, as well as cash and in-kind contribution from the industrial partner, Rolls-Royce.
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Long Ma is an associate researcher at the School of Software, Shandong University. He holds his M.S. degrees in mathematics and in computer science from Shandong University. His research focuses on computer graphics, geometry modeling, and mechanical simulation.
Sidan Yao is a Ph.D. student in the School of Computer Science and Engineering, Nanyang Technological University. She received her bachelor degree from the School of Computer Software, Tianjin University. Her research focuses on architectural geometry and differential geometry.
Jianmin Zheng is a full professor in the School of Computer Engineering at Nanyang Technological University. He received his bachelor and Ph.D. degrees from Zhejiang University. His research interests include computer aided geometric design, computer graphics, geometric modeling, CAD, visualization, and interactive digital media.
Yang Liu is a principal researcher in the Internet Graphics Group at Microsoft Research Asia which he joined in 2010. He received his Ph.D. degree in computer science from the University of Hong Kong in 2008, and his master and bachelor degrees in computational mathematics from the University of Science and Technology of China, in 2003 and 2000 respectively. He worked in the Alice group at INRIA/LORIA as a post-doctoral researcher from 2008. His research interests span geometric modeling and optimization, mesh generation, computer-aided geometric design, and architectural geometry.
Yuanfeng Zhou is a professor in the School of Software, Shandong University. He received his B.S., M.S., and Ph.D. degrees from Shandong University. His research focuses on intelligent graphics and image processing, geometric modeling and optimization, computational medicine, and virtual reality.
Shi-Qing Xin is an associate professor at the School of Computer Science and Technology, Shandong University. He received his Ph.D. degree in applied mathematics from Zhejiang University. His research focuses on geometric calculation, geometric modeling, and scene understanding.
Ying He is an associate professor in the School of Computer Engineering, Nanyang Technological University. He received his B.S. and M.S. degrees in electrical engineering from Tsinghua University, and his Ph.D. degree in computer science from Stony Brook University. His research interests fall into the general areas of visual computing and he is particularly interested in problems which require geometric analysis and computation.
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Ma, L., Yao, S., Zheng, J. et al. Constructing self-supporting surfaces with planar quadrilateral elements. Comp. Visual Media 8, 571–583 (2022). https://doi.org/10.1007/s41095-021-0257-1
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DOI: https://doi.org/10.1007/s41095-021-0257-1