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A proof of Pohlke’s theorem with an analytic determination of the reference trihedron. (English) Zbl 1415.51032
The author offers a linear algebra proof of Pohlke’s theorem (Given three arbitrary segments \(OP_1\), \(OP_2\), \(OP_3\) in a plane, not contained in a line, there is a cube, such that the parallel projection of three of its edges \(OQ_1\), \(OQ_2\), \(QQ_3\) is \(OP_1\), \(OP_2\), \(OP_3\)) and provides “explicit formulae for the reference trihedrons (Pohlke matrices) and the corresponding directions of projection.”

51N10 Affine analytic geometry
51N05 Descriptive geometry
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[1] N. Beskin: An Analogue of Pohlke-Schwarz’s theorem in central axonometry. Recueil Mathématique [Math. Sbornik] N.S., 19 (61), 5772 (1946). · Zbl 0061.38502
[2] H. Brauner: Lineare Abbildungen aus Euklidischen Räumen. Beitr. Algebra Geom. 21, 526 (1986). · Zbl 0589.51004
[3] L. Campedelli: Lezioni di Geometria. Vol. II, Parte I, Cedam, Padova 1972. · Zbl 0046.38102
[4] A. Cayley: On a Problem of Projection. Quartely Journal of Pure and Applied Mathematics XIII, 1929 (1875).
[5] A. Emch: Proof of Pohlke’s Theorem and Its Generalizations by Anity. Amer. J. Math. 40, 366374 (1918).
[6] H. Eves: Elementary Matrix Theory. Dover Publications, New York 1966.
[7] F. Klein: Elementary Mathematics From An Advanced Standpoint. Geometry. Dover Publications, New York 1939. · JFM 65.0640.01
[8] G. Loria: Storia della Geometria Descrittiva. Hoepli, Milano 1921.
[9] C.F. Manara: L’aspetto Algebrico di un Fondamentale Teorema di Geometria Descrittiva. Periodico di Matematiche  Serie IV, XXXII, 142149 (1954).
[10] E. Müller, E. Kruppa: Vorlesungen über Darstellende Geometrie, I. Band: Die Linearen Abbildungen. Franz Deuticke, Leipzig und Wien 1923, pp. 172181.
[11] K.W. Pohlke: Lehrbuch der Darstellenden Geometrie. Part I, Berlin 1860.
[12] H.A. Schwarz: Elementarer Beweis des Pohlkeschen Fundamentalsatzes der Axonometrie. Crelle’s Journal LXIII, 309314 (1864).
[13] H. Stachel: Mehrdimensionale Axonometrie. In N.K. Stephanidis (ed.): Proceedings of the Congress of Geometry, Thessaloniki 1987, pp. 159168.
[14] H. Steinhaus: Mathematical Snapshots. Oxford University Press, Oxford 1950. · Zbl 0041.27502
[15] E. Stiefel: Zum Satz von Pohlke. Comment. Math. Helv. 10, 208225 (1938). · JFM 64.0649.02
[16] D.J. Struik: Lectures on Analytic and Projective Geometry. Addison-Wesley, 1953. Received March 29, 2018; nal form November 17, 2018
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