{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:53:26Z","timestamp":1777424006039,"version":"3.51.4"},"reference-count":19,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2016,12,1]],"date-time":"2016-12-01T00:00:00Z","timestamp":1480550400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>The real projective plane has been formalized in Isabelle\/HOL by Timothy Makarios [13] and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck [12].<\/jats:p>\n               <jats:p>Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk [9], Krzysztof Prazmowski [10] and by Wojciech Skaba [18].<\/jats:p>\n               <jats:p>In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Pl\u00fccker relation in rank 3 [2], [1], [7], [16], [17].<\/jats:p>\n               <jats:p>Then we show that the projective space induced (in the sense defined in [9]) by \u211d<jats:sup>3<\/jats:sup> is a projective plane (in the sense defined in [10]).<\/jats:p>\n               <jats:p>Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear.<\/jats:p>","DOI":"10.1515\/forma-2016-0020","type":"journal-article","created":{"date-parts":[[2017,2,25]],"date-time":"2017-02-25T10:00:53Z","timestamp":1488016853000},"page":"239-251","source":"Crossref","is-referenced-by-count":3,"title":["Homography in \u211d\u2119"],"prefix":"10.1515","volume":"24","author":[{"given":"Roland","family":"Coghetto","sequence":"first","affiliation":[{"name":"Rue de la Brasserie 5, 7100 La Louvi\u00e8re, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,2,23]]},"reference":[{"key":"2021040802442562182_j_forma-2016-0020_ref_001_w2aab2b8c14b1b7b1ab1ab1Aa","unstructured":"[1] Susanne Apel. The geometry of brackets and the area principle. Phd thesis, Technische Universit\u00e4t M\u00fcnchen, Fakult\u00e4t f\u00fcr Mathematik, 2014."},{"key":"2021040802442562182_j_forma-2016-0020_ref_002_w2aab2b8c14b1b7b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Susanne Apel and J\u00fcrgen Richter-Gebert. Cancellation patterns in automatic geometric theorem proving. In Automated Deduction in Geometry, pages 1\u201333. Springer, 2010.","DOI":"10.1007\/978-3-642-25070-5_1"},{"key":"2021040802442562182_j_forma-2016-0020_ref_003_w2aab2b8c14b1b7b1ab1ab3Aa","unstructured":"[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107\u2013114, 1990."},{"key":"2021040802442562182_j_forma-2016-0020_ref_004_w2aab2b8c14b1b7b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2021040802442562182_j_forma-2016-0020_ref_005_w2aab2b8c14b1b7b1ab1ab5Aa","unstructured":"[5] Czes\u0142aw Byli\u0144ski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529\u2013536, 1990."},{"key":"2021040802442562182_j_forma-2016-0020_ref_006_w2aab2b8c14b1b7b1ab1ab6Aa","unstructured":"[6] Agata Darmochwa\u0142. The Euclidean space. Formalized Mathematics, 2(4):599\u2013603, 1991."},{"key":"2021040802442562182_j_forma-2016-0020_ref_007_w2aab2b8c14b1b7b1ab1ab7Aa","doi-asserted-by":"crossref","unstructured":"[7] Laurent Fuchs and Laurent Thery. A formalization of Grassmann-Cayley algebra in Coq and its application to theorem proving in projective geometry. In Automated Deduction in Geometry, pages 51\u201367. Springer, 2010.","DOI":"10.1007\/978-3-642-25070-5_3"},{"key":"2021040802442562182_j_forma-2016-0020_ref_008_w2aab2b8c14b1b7b1ab1ab8Aa","unstructured":"[8] Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381\u2013383, 2003."},{"key":"2021040802442562182_j_forma-2016-0020_ref_009_w2aab2b8c14b1b7b1ab1ab9Aa","unstructured":"[9] Wojciech Leo\u0144czuk and Krzysztof Pra\u017cmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761\u2013766, 1990."},{"key":"2021040802442562182_j_forma-2016-0020_ref_010_w2aab2b8c14b1b7b1ab1ac10Aa","unstructured":"[10] Wojciech Leo\u0144czuk and Krzysztof Pra\u017cmowski. Projective spaces \u2013 part I. Formalized Mathematics, 1(4):767\u2013776, 1990."},{"key":"2021040802442562182_j_forma-2016-0020_ref_011_w2aab2b8c14b1b7b1ab1ac11Aa","doi-asserted-by":"crossref","unstructured":"[11] Xiquan Liang, Piqing Zhao, and Ou Bai. Vector functions and their differentiation formulas in 3-dimensional Euclidean spaces. 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