Group of Homography in Real Projective Plane

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Using the Mizar system [2], we formalized that homographies of the projective real plane (as defined in [5]), form a group.

Then, we prove that, using the notations of Borsuk and Szmielew in [3]

“Consider in space ℝℙ2 points P1, P2, P3, P4 of which three points are not collinear and points Q1,Q2,Q3,Q4 each three points of which are also not collinear. There exists one homography h of space ℝℙ2 such that h(Pi) = Qi for i = 1, 2, 3, 4.”

(Existence Statement 52 and Existence Statement 53) [3]. Or, using notations of Richter [11]

“Let [a], [b], [c], [d] in ℝℙ2 be four points of which no three are collinear and let [a′],[b′],[c′],[d′] in ℝℙ2 be another four points of which no three are collinear, then there exists a 3 × 3 matrix M such that [Ma] = [a′], [Mb] = [b′], [Mc] = [c′], and [Md] = [d′]”

Makarios has formalized the same results in Isabelle/Isar (the collineations form a group, lemma statement52-existence and lemma statement 53-existence) and published it in Archive of Formal Proofs [10], [9].

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[2] Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-8 17.

[3] Karol Borsuk and Wanda Szmielew. Foundations of Geometry. North Holland, 1960.

[4] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.

[5] Roland Coghetto. Homography in RP2. Formalized Mathematics, 24(4):239-251, 2016. doi: 10.1515/forma-2016-0020.

[6] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.

[7] Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381-383, 2003.

[8] Wojciech Leończuk and Krzysztof Prazmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761-766, 1990.

[9] Timothy James McKenzie Makarios. The independence of Tarski’s Euclidean Axiom. Archive of Formal Proofs, October 2012.

[10] Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University of Wellington, New Zealand, 2012. Master’s thesis.

[11] Jürgen Richter-Gebert. Perspectives on projective geometry: a guided tour through real and complex geometry. Springer Science & Business Media, 2011.

Formalized Mathematics

(a computer assisted approach)

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researchers in the fields of formal methods and computer-checked mathematics

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