{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,19]],"date-time":"2025-03-19T10:50:56Z","timestamp":1742381456341},"reference-count":41,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2013,3,1]],"date-time":"2013-03-01T00:00:00Z","timestamp":1362096000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Form. Asp. Comput."],"published-print":{"date-parts":[[2013,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A spatial object consists of data assigned to points in a space. Spatial objects, such as memory states and three dimensional graphical scenes, are diverse and ubiquitous in computing. We develop a general theory of spatial objects by modelling abstract data types of spatial objects as topological algebras of functions. One useful algebra is that of continuous functions, with operations derived from operations on space and data, and equipped with the compact-open topology. Terms are used as abstract syntax for defining spatial objects and conditional equational specifications are used for reasoning. We pose a completeness problem:<jats:italic>Given a selection of operations on spatial objects, do the terms approximate all the spatial objects to arbitrary accuracy?<\/jats:italic>We give some general methods for solving the problem and consider their application to spatial objects with real number attributes.<\/jats:p>","DOI":"10.1007\/s00165-011-0182-7","type":"journal-article","created":{"date-parts":[[2011,7,18]],"date-time":"2011-07-18T07:26:14Z","timestamp":1310973974000},"page":"189-218","source":"Crossref","is-referenced-by-count":2,"title":["The data type of spatial objects"],"prefix":"10.1145","volume":"25","author":[{"given":"Kenneth","family":"Johnson","sequence":"first","affiliation":[{"name":"INRIA Rennes-Bretagne Atlantique, Campus Universitaire de Beaulieu, 35042, Rennes, France"},{"name":"Aston University, Aston Triangle, B4 7ET, Birmingham, UK"}]},{"given":"John V.","family":"Tucker","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, UK"}]}],"member":"320","reference":[{"key":"e_1_2_1_2_1_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-8659.2005.00866.x"},{"key":"e_1_2_1_2_2_2","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz M","year":"1964"},{"key":"e_1_2_1_2_3_2","unstructured":"Bourbaki N (1966) General topology part 1 and 2. Springer"},{"key":"e_1_2_1_2_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00065-2"},{"key":"e_1_2_1_2_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90123-X"},{"key":"e_1_2_1_2_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/227683.227687"},{"key":"e_1_2_1_2_7_2","unstructured":"Chen M Tucker JV (1998) Constructive volume geometry. Technical report CS-TR-98-19 University of Wales Swansea July 1998"},{"key":"e_1_2_1_2_8_2","doi-asserted-by":"publisher","DOI":"10.1111\/1467-8659.00464"},{"key":"e_1_2_1_2_9_2","unstructured":"Dugundji J (1966) Topology. Allyn and Bacon Inc."},{"key":"e_1_2_1_2_10_2","doi-asserted-by":"crossref","unstructured":"Edalat A Lieutier A (1999) Foundation of a computable solid modeling. In: Proceedings of the fifth ACM symposium on solid modeling and applications pp 278\u2013284","DOI":"10.1145\/304012.304040"},{"key":"e_1_2_1_2_11_2","unstructured":"Engelking R (1989) General topology. Heldermann Verlag"},{"key":"e_1_2_1_2_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/PL00003936"},{"key":"e_1_2_1_2_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S1567-8326(03)00041-9"},{"key":"e_1_2_1_2_14_2","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-77487-9","volume-title":"Universal algebra","author":"Gr\u00e4tzer G","year":"1979","edition":"2"},{"key":"e_1_2_1_2_15_2","unstructured":"Hearn D Baker MP (2004) Computer graphics with OpenGL 3rd edn. Pearson Prentice-Hall"},{"key":"e_1_2_1_2_16_2","volume-title":"Analysis in Euclidean space","author":"Hoffman K","year":"1975"},{"key":"e_1_2_1_2_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2007.48"},{"key":"e_1_2_1_2_18_2","unstructured":"Johnson K (2007) The algebraic specification of spatial data types with applications to constructive volume geometry. PhD thesis University of Wales Swansea"},{"key":"e_1_2_1_2_19_2","volume-title":"General topology","author":"Kelley JL","year":"1975"},{"key":"e_1_2_1_2_20_2","first-page":"593","article-title":"Ein beitrag zur optik der farbanstriche","volume":"12","author":"Kubelka P","year":"1931","journal-title":"Zeitschrift f\u00fcr Technische Physik"},{"key":"e_1_2_1_2_21_2","doi-asserted-by":"publisher","DOI":"10.1109\/38.511"},{"key":"e_1_2_1_2_22_2","doi-asserted-by":"crossref","unstructured":"McCoy NH (1948) Rings and ideals. The Mathematical Association of America","DOI":"10.5948\/9781614440086"},{"key":"e_1_2_1_2_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(95)00261-8"},{"key":"e_1_2_1_2_24_2","first-page":"189","volume-title":"HOA \u201993: selected papers from the first international workshop on higher-order algebra, logic, and term rewriting, vol 816","author":"Meinke K","year":"1993"},{"key":"e_1_2_1_2_25_2","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1093\/oso\/9780198537359.003.0003","volume-title":"Handbook of logic for computer science","author":"Meinke K","year":"1992"},{"key":"e_1_2_1_2_26_2","volume-title":"Topological methods in Euclidean spaces","author":"Naber GL","year":"1980"},{"key":"e_1_2_1_2_27_2","doi-asserted-by":"crossref","unstructured":"Porter T Duff T (1984) Compositing digital images. In: SIGGRAPH \u201984: proceedings of the 11th annual conference on computer graphics and interactive techniques. ACM Press New York pp 253\u2013259","DOI":"10.1145\/800031.808606"},{"key":"e_1_2_1_2_28_2","doi-asserted-by":"publisher","DOI":"10.1145\/356827.356833"},{"key":"e_1_2_1_2_29_2","doi-asserted-by":"publisher","DOI":"10.1109\/MCG.1982.1674149"},{"key":"e_1_2_1_2_30_2","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1093\/oso\/9780198537809.003.0004","volume-title":"Handbook of logic for computer science, vol 4","author":"Stoltenberg-Hansen V","year":"1995"},{"key":"e_1_2_1_2_31_2","volume-title":"Introduction to topology and modern analysis","author":"Simmons GF","year":"1963"},{"key":"e_1_2_1_2_32_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00371-006-0028-0"},{"key":"e_1_2_1_2_33_2","doi-asserted-by":"publisher","DOI":"10.2307\/3029750"},{"key":"e_1_2_1_2_34_2","unstructured":"Stroud KA (2007) Engineering mathematics 6th edn. Palgrave Macmillan"},{"key":"e_1_2_1_2_35_2","unstructured":"Stephenson K Tucker JV (2011) Data syntax and semantics an introduction to modelling programming languages. Swansea University"},{"key":"e_1_2_1_2_36_2","first-page":"317","volume-title":"Handbook of logic for computer science, vol 5","author":"Tucker JV","year":"1998"},{"key":"e_1_2_1_2_37_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00297-7"},{"key":"e_1_2_1_2_38_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.scico.2005.09.001"},{"key":"e_1_2_1_2_39_2","doi-asserted-by":"crossref","unstructured":"Winter AS Chen M (2001) vlib : a volume graphics API. In: Volume graphics 2001. Springer-Wien New York","DOI":"10.1007\/978-3-7091-6756-4_9"},{"key":"e_1_2_1_2_40_2","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-76771-5","volume-title":"Universal algebra for computer scientists","author":"Wechler W","year":"1992"},{"key":"e_1_2_1_2_41_2","unstructured":"Winter AS. http:\/\/vg.swan.ac.uk\/vlib\/. Accessed July 2011"}],"container-title":["Formal Aspects of Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00165-011-0182-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00165-011-0182-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1007\/s00165-011-0182-7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T19:50:35Z","timestamp":1712605835000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1007\/s00165-011-0182-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,3]]}},"alternative-id":["10.1007\/s00165-011-0182-7"],"URL":"https:\/\/doi.org\/10.1007\/s00165-011-0182-7","relation":{},"ISSN":["0934-5043","1433-299X"],"issn-type":[{"value":"0934-5043","type":"print"},{"value":"1433-299X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3]]}}}