Homography in ℝℙ

Roland Coghetto

Formalized Mathematics (2016)

  • Volume: 24, Issue: 4, page 239-251
  • ISSN: 1426-2630

Abstract

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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]. 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]. In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17]. Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]). 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.

How to cite

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Roland Coghetto. "Homography in ℝℙ." Formalized Mathematics 24.4 (2016): 239-251. <http://eudml.org/doc/287998>.

@article{RolandCoghetto2016,
abstract = {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]. 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]. In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17]. Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]). 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.},
author = {Roland Coghetto},
journal = {Formalized Mathematics},
keywords = {projectivity; projective transformation; projective collineation; real projective plane; Grassmann-Plücker relation},
language = {eng},
number = {4},
pages = {239-251},
title = {Homography in ℝℙ},
url = {http://eudml.org/doc/287998},
volume = {24},
year = {2016},
}

TY - JOUR
AU - Roland Coghetto
TI - Homography in ℝℙ
JO - Formalized Mathematics
PY - 2016
VL - 24
IS - 4
SP - 239
EP - 251
AB - 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]. 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]. In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17]. Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]). 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.
LA - eng
KW - projectivity; projective transformation; projective collineation; real projective plane; Grassmann-Plücker relation
UR - http://eudml.org/doc/287998
ER -

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