{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:34:52Z","timestamp":1753889692976,"version":"3.41.2"},"reference-count":1,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2012,9,13]],"date-time":"2012-09-13T00:00:00Z","timestamp":1347494400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>In mathematics curves are typically defined as the images of continuous real\nfunctions (parametrizations) defined on a closed interval. They can also be\ndefined as connected one-dimensional compact subsets of points. For simple\ncurves of finite lengths, parametrizations can be further required to be\ninjective or even length-normalized. All of these four approaches to curves are\nclassically equivalent. In this paper we investigate four different versions of\ncomputable curves based on these four approaches. It turns out that they are\nall different, and hence, we get four different classes of computable curves.\nMore interestingly, these four classes are even point-separable in the sense\nthat the sets of points covered by computable curves of different versions are\nalso different. However, if we consider only computable curves of computable\nlengths, then all four versions of computable curves become equivalent. This\nshows that the definition of computable curves is robust, at least for those of\ncomputable lengths. In addition, we show that the class of computable curves of\ncomputable lengths is point-separable from the other four classes of computable\ncurves.<\/jats:p>","DOI":"10.2168\/lmcs-8(3:15)2012","type":"journal-article","created":{"date-parts":[[2013,11,29]],"date-time":"2013-11-29T08:17:46Z","timestamp":1385713066000},"source":"Crossref","is-referenced-by-count":2,"title":["Point-Separable Classes of Simple Computable Planar Curves"],"prefix":"10.46298","volume":"Volume 8, Issue 3","author":[{"given":"Xizhong","family":"Zheng","sequence":"first","affiliation":[]},{"given":"Robert","family":"Rettinger","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2012,9,13]]},"reference":[{"key":"699:not-found"}],"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/941\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/941\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T19:59:31Z","timestamp":1681243171000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/941"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,13]]},"references-count":1,"URL":"https:\/\/doi.org\/10.2168\/lmcs-8(3:15)2012","relation":{"is-same-as":[{"id-type":"arxiv","id":"1208.2245","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1208.2245","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2012,9,13]]},"article-number":"941"}}