Abstract
In this paper, we compare the performance of three different folding models when they are applied to three different map folding settings. Precisely, the three folding models include the simple folding model, the simple folding–unfolding model, and the general folding model. The different map folding settings are discussed by comparing different folded states, i.e., different overlapping orders on the set of the squares of 1 × n maps, the squares of m × n maps, and the squares lying on the boundary of m × n maps. These folding models are abstracts of manual works and robotics. We clarify the relationship between their reachable final folded states under different settings and give proof of all the inclusion relationships between every two of these sets. In addition, there are nine distinct problems with the three folding models applied to three folding settings. We give the optimal linear time solutions to all the unsolved ones: the valid total overlapping order problems of 1 × n maps, m × n maps, as well as the valid boundary overlapping order problems of m × n maps with the three different folding models. Our work gives the conclusion of the research field where the folding models and the overlapping orders of map folding are concerned.
Highlights
Problems in the field of origami are becoming more and more popular because of their applicability in robotics and computational modeling, especially the origamis whose crease patterns are relatively simple
In [15], validity of given overlapping orders (VOP) considering the total order of m × n squares in an m × n map with the general folding model is proven to be linear-time solvable
In [16,17], two linear-time algorithms are proposed to decide: (1) the validity of overlapping orders given on all the squares of an m × n map (Total VOP) using the simple folding–unfolding model; (2) the validity of boundary overlapping orders of an m × n map (Boundary VOP) using the simple folding model, respectively
Summary
Problems in the field of origami are becoming more and more popular because of their applicability in robotics and computational modeling, especially the origamis whose crease patterns (comprising all the folding line segments and their intersections) are relatively simple. In [15], VOP considering the total order of m × n squares in an m × n map with the general folding model is proven to be linear-time solvable. Such a result does not give enough hint on the computational complexity of the map folding problem because a flat-foldable map may have exponential flat-folded states. In [16,17], two linear-time algorithms are proposed to decide: (1) the validity of overlapping orders given on all the squares of an m × n map (Total VOP) using the simple folding–unfolding model; (2) the validity of boundary overlapping orders of an m × n map (Boundary VOP) using the simple folding model, respectively. Even if we restrict the overlapping orders to only the boundary squares, the strict inclusion relation still keeps (i.e., Boundary VOP)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
