{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T09:46:27Z","timestamp":1777715187859,"version":"3.51.4"},"reference-count":14,"publisher":"SAGE Publications","issue":"5","license":[{"start":{"date-parts":[[1991,10,1]],"date-time":"1991-10-01T00:00:00Z","timestamp":686275200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The International Journal of Robotics Research"],"published-print":{"date-parts":[[1991,10]]},"abstract":"<jats:p>\n                    In this article we propose a modeling space for objects defined in terms of tolerances based on Requicha's sug gestion of using variational classes. Variational classes are subsets of the hyperspace 2\n                    <jats:sup>En<\/jats:sup>\n                    , where E\n                    <jats:sup>n<\/jats:sup>\n                    is euclidean n-space. In order to motivate the ideas, the discussion involves the same simple example throughout: the specifi cation of a ball bearing defined by position, size, and form constraints. We begin by discussing the relationship between Reguicha's original proposal and our proposal for a definition of what should be viewed as a permissible variational class. We call such a permissible class, together with a nominal solid S, an R-class. We then introduce generalized versions of the regularized Boolean operations, which operate not on r-sets, but rather on R- classes. Just as the r-sets are closed under regularized Boolean operations, so the R-classes are closed under the generalized versions of the regularized Boolean opera tions. Finally, we discuss the relationship between the R- classes and the particular feature tolerancing methods proposed by Requicha.\n                  <\/jats:p>","DOI":"10.1177\/027836499101000512","type":"journal-article","created":{"date-parts":[[2007,3,11]],"date-time":"2007-03-11T03:05:13Z","timestamp":1173582313000},"page":"570-582","source":"Crossref","is-referenced-by-count":49,"title":["Modeling Spaces for Toleranced Objects"],"prefix":"10.1177","volume":"10","author":[{"given":"M.","family":"Boyer","sequence":"first","affiliation":[{"name":"D\u00e9partement d'informatique et de recherche op\u00e9rationnelle Universit\u00e9 de Montr\u00e9al Montr\u00e9al, Qu\u00e9bec, Canada H3C 3J7"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N.F.","family":"Stewart","sequence":"additional","affiliation":[{"name":"D\u00e9partement d'informatique et de recherche op\u00e9rationnelle Universit\u00e9 de Montr\u00e9al Montr\u00e9al, Qu\u00e9bec, Canada H3C 3J7"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[1991,10,1]]},"reference":[{"key":"atypb1","volume-title":"An extension of manifold boundary representations to the r-sets","author":"Desaulniers, H.","year":"1988"},{"key":"atypb2","unstructured":"Fleming, A.D. 1989. A representation for geometrically toleranced parts. In Woodwark, J. (ed.): Geometric Reasoning Oxford, England: Oxford University Press, pp. 141-168."},{"key":"atypb3","volume-title":"Topology","author":"Hocking, J.G.","year":"1961"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1134-1"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1145\/15886.15904"},{"issue":"18","key":"atypb6","first-page":"449","author":"Lojasiewicz, S.","year":"1964","journal-title":"Annali Scienze, Fisiche e Mathematiche"},{"key":"atypb7","volume-title":"An Introduction to Solid Modeling","author":"M\u00e4ntyl\u00e4, M.","year":"1988"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0146-664X(82)90104-6"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1951-0042109-4"},{"key":"atypb10","volume-title":"Hyperspaces of Sets","author":"Nadler, S.B.","year":"1978"},{"key":"atypb11","volume-title":"Mathematical models of rigid solid objects. Tech. memo. no. 28, Production Automation Project","author":"Requicha, A.A.G.","year":"1977"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1177\/027836498300200403"},{"key":"atypb13","volume-title":"Representation of tolerances in solid modeling: Issues and alternative approaches. Pickett, M. S., and Boyse, J. W. (eds.): Solid Modeling by Computers","author":"Requicha, A.A.G.","year":"1984"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1145\/37402.37421"}],"container-title":["The International Journal of Robotics Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/027836499101000512","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/027836499101000512","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T10:14:20Z","timestamp":1777457660000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/027836499101000512"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,10]]},"references-count":14,"journal-issue":{"issue":"5","published-print":{"date-parts":[[1991,10]]}},"alternative-id":["10.1177\/027836499101000512"],"URL":"https:\/\/doi.org\/10.1177\/027836499101000512","relation":{},"ISSN":["0278-3649","1741-3176"],"issn-type":[{"value":"0278-3649","type":"print"},{"value":"1741-3176","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,10]]}}}