Evolutionary Piecewise Developable Approximations
ZhengYu Zhao* Mo Li* Zheng Zhang Qing Fang Ligang Liu XiaoMing Fu  
(*Joint first authors)  
University of Science and Technology of China 
ACM Transactions on Graphics (Proc. SIGGRAPH), 42(4), 2023.
Teaser: Our algorithm generates a small number of developable patches for approximating three models, including a model with three mushrooms, a chair, and a teapot, physically validated using color paper. 

Abstract 

Keywords 
Piecewise developable approximation, evolutionary genetic algorithm, conformal mapping, cone singularity 
Motivation 

Methods 
Figure 2: The generation of one chromosome. (a) The input. (b) The set of cone singularities. (c) The minimum spanning tree connecting all cone singularities. (d) The initial partition whose boundary contains this minimum spanning tree. Then it performs genetic operations to evolve the population for finding a chromosome with a small fitness value until the termination conditions are satisfied.
Figure 3: Workflow of our method. Given an input mesh M (a), our algorithm first generates an initial population A (the best chromosome in the initial population is shown in (b)), then evolves the population for finding a chromosome with a small fitness value (c), and finally deform patches to be piecewise developable to generate the result (d). The graph plots the fitness of the best chromosome P best in the population as a function of the number of evolutionary iterations. The red, orange, pink, blue, green, and purple numbers denote E_{total}(P_{best}), E_{small}(P_{best}), E_{narrow}(P_{best}), E_{length}(P_{best}), E_{smooth}(P_{best}), and ε(P_{best}), respectively. The sudden drop of the fitness function near the 200^{th} iteration is due to the disappearance of the narrow regions.
Specifically, the operations include selection, crossover, and mutation.
Given the best partition, we deform the input mesh to be piecewise developable while possessing a small approximation error.

Paper 

Software 
Code (15.3M) 
Supplementary material 
Supplement and Data 
Video 
Demo (*.mp4) (78.6M) 
Ack 

BibTex 
@article {zhao2023evolutionary, title = {Evolutionary Piecewise Developable Approximations}, author = {Zhao, ZhengYu and Li, Mo and Zhang, Zheng and Fang, Qing and Liu, Ligang and Fu, XiaoMing} journal = {ACM Transactions on Graphics}, volume={42}, number={4}, pages={113}, year = {2023} } 
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