Homography in \(\mathbb{R}\mathbb{P}^2\). (English) Zbl 1357.51021
Summary: The real projective plane has been formalized in Isabelle/HOL by T. J. McKenzie Makarios [A mechanical verification of the independence of Tarski’s Euclidean axiom. Wellington: Victoria University of Wellington (PhD Thesis) (2012)] and in Coq by N. Magaud et al. [Lect. Notes Comput. Sci. 6301, 141–162 (2011; Zbl 1302.68246)].
Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by W. Leonczuk and K. Prazmowski [“A construction of analytical projective space”, Formaliz. Math. 1, No. 4, 761–766 (1990), “Projective spaces. I”, ibid. 1, No. 4, 767–776 (1990)] and by W. Skaba [“The collinearity structure”, ibid. 1, No. 4, 657–659 (1990)].
In this article, we check with the Mizar system, some properties on the determinants and the Grassmann-Plücker relation in rank 3.
Then we show that the projective space induced (in the sense defined in [Leonczuk and Prazmowski, “A construction of analytical projective space”, loc. cit.]) by \(\mathbb{R}^3\) is a projective plane (in the sense defined in [Leonczuk and Prazmowsk, “‘Projective spaces. I”, loc. cit.]).
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.
Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by W. Leonczuk and K. Prazmowski [“A construction of analytical projective space”, Formaliz. Math. 1, No. 4, 761–766 (1990), “Projective spaces. I”, ibid. 1, No. 4, 767–776 (1990)] and by W. Skaba [“The collinearity structure”, ibid. 1, No. 4, 657–659 (1990)].
In this article, we check with the Mizar system, some properties on the determinants and the Grassmann-Plücker relation in rank 3.
Then we show that the projective space induced (in the sense defined in [Leonczuk and Prazmowski, “A construction of analytical projective space”, loc. cit.]) by \(\mathbb{R}^3\) is a projective plane (in the sense defined in [Leonczuk and Prazmowsk, “‘Projective spaces. I”, loc. cit.]).
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.
Keywords:
projectivity; projective transformation; projective collineation; real projective plane; Grassmann-Plücker relationCitations:
Zbl 1302.68246References:
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