{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:12:49Z","timestamp":1760148769538,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T00:00:00Z","timestamp":1685491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004396","name":"Thailand Research Fund","doi-asserted-by":"publisher","award":["RSA-6180074"],"award-info":[{"award-number":["RSA-6180074"]}],"id":[{"id":"10.13039\/501100004396","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We consider a problem in computational origami. Given a piece of paper as a convex polygon P and a point f located within, we fold every point on a boundary of P to f and compute a region that is safe from folding, i.e., the region with no creases. This problem is an extended version of a problem by Akitaya, Ballinger, Demaine, Hull, and Schmidt that only folds corners of the polygon. To find the region, we prove structural properties of intersections of parabola-bounded regions and use them to devise a linear-time algorithm. We also prove a structural result regarding the complexity of the safe region as a variable of the location of point f, i.e., the number of arcs of the safe region can be determined using the straight skeleton of the polygon P.<\/jats:p>","DOI":"10.3390\/a16060281","type":"journal-article","created":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T02:12:45Z","timestamp":1685585565000},"page":"281","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Folding Every Point on a Polygon Boundary to a Point"],"prefix":"10.3390","volume":"16","author":[{"given":"Nattawut","family":"Phetmak","sequence":"first","affiliation":[{"name":"Faculty of Computer Engineering, Kasetsart University, Bangkok 10900, Thailand"}]},{"given":"Jittat","family":"Fakcharoenphol","sequence":"additional","affiliation":[{"name":"Faculty of Computer Engineering, Kasetsart University, Bangkok 10900, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lang, R.J. (2009, January 8\u201310). Computational Origami: From Flapping Birds to Space Telescopes. Proceedings of the Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry (SCG \u201909), Aarhus, Denmark.","DOI":"10.1145\/1542362.1542363"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Demaine, E.D., and O\u2019Rourke, J. (2008). Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press. reprint ed.","DOI":"10.1017\/CBO9780511735172"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hull, T.C. (2020). Origametry: Mathematical Methods in Paper Folding, Cambridge University Press.","DOI":"10.1017\/9781108778633"},{"key":"ref_4","unstructured":"Akitaya, H.A., Ballinger, B., Demaine, E.D., Hull, T.C., and Schmidt, C. (2021, January 10\u201312). Folding Points to a Point and Lines to a Line. 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