Pohlke's theorem

The set of Pohlke , even fundamental theorem of axonometry or law of Axonometry called, is a tenet of mathematical sub-region of Descriptive Geometry . It goes back to Karl Wilhelm Pohlke and deals with a fundamental question of axonometry .

Formulation of the sentence

Pohlke's theorem

The sentence can be summarized as follows:

(P) Any planar tripod of the Euclidean space , the distances not all on a straight line lying can be seen as defined by a parallel projection resulting image of orthonormal spatial tripod . ${\displaystyle \mathbb {R} ^{3}}$

In a more general way:

(P ^' ) Three in a level of from a given point outgoing routes of arbitrary length and arbitrary direction can be regarded as a parallel projection of three in a further given point abutting edges of the cube is as long provided that at most three of the first-mentioned points are collinear. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle {\overline {O}}}$ ${\displaystyle O}$

In general:

(PS) are in three-dimensional Euclidean space one level and also two points and given and go from the former three arbitrary lines which, despite a common vertex , however, have no common plane are , ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle E\subset \mathbb {R} ^{3}}$ ${\displaystyle O\in \mathbb {R} ^{3}}$ ${\displaystyle {\overline {O}}\in E}$ ${\displaystyle S_{1},S_{2},S_{3}\subset \mathbb {R} ^{3}}$ ${\displaystyle O}$

while from the latter three further arbitrary routes start, which have a common corner point, but - although lying in the plane - are not collinear , ${\displaystyle {\overline {S_{1}}},{\overline {S_{2}}},{\overline {S_{3}}}\subset E}$ ${\displaystyle {\overline {O}}}$ ${\displaystyle E}$

so there is always

a similitude map as well ${\displaystyle \eta \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}}$

a movement of space and finally ${\displaystyle \beta \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}}$

a parallel projection , ${\displaystyle \phi \colon \mathbb {R} ^{3}\rightarrow E}$

so that the linked mapping maps the corner point to the other corner point and thereby to . ${\displaystyle \psi =\phi \circ \beta \circ \eta }$ ${\displaystyle O}$ ${\displaystyle {\overline {O}}=\psi (O)}$ ${\displaystyle S_{k}}$ ${\displaystyle {\overline {S_{k}}}=\psi (S_{k})\;(k=1,2,3)}$

Notes on the history of the sentence

Pohlke found the fundamental theorem around 1853. His original proof was extraordinarily complicated and remained unpublished . Hermann Amandus Schwarz , who was a pupil of Pohlke, published the first complete proof in 1864 and also provided the more general representation presented above (PS) . The fundamental theorem - and representations equivalent to it - are therefore also referred to by some authors as theorem by Pohlke and Schwarz ( English Pohlke-Schwarz theorem ).

Corollary

The following corollary can be obtained from the fundamental theorem, which can be regarded as equivalent in terms of its expressiveness:

(PS ^' ) Every complete quadrangle lying in one plane can be understood as an image of a tetrahedron , which is created by parallel projection , which is similar to a given tetrahedron . ${\displaystyle E\subset \mathbb {R} ^{3}}$ ${\displaystyle {\mathcal {V}}\subset E}$ ${\displaystyle {\mathcal {T}}^{*}\subset \mathbb {R} ^{3}}$ ${\displaystyle {\mathcal {T}}\subset \mathbb {R} ^{3}}$

literature

• PS Alexandroff , AI Markuschewitsch , AJ Chintschin : Encyclopedia of Elementary Mathematics . Volume IV. Geometry (=  university books for mathematics . Volume 10 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1969, p. 250-254 .
• Heinrich Brauner : Textbook of constructive geometry . Springer Verlag, Vienna / New York 1969, ISBN 3-211-81833-2 , pp. 51, 85-86 .
• Hermann Athens, Jörn Bruhn (ed.): Lexicon of school mathematics . and adjacent areas. 5th edition. tape 3 . LR. Aulis Verlag Deubner, Cologne 1977, ISBN 3-7614-0242-2 , p. 775 .
• Wolfgang Haack : Descriptive Geometry . Volume III: Axonometry and Perspective (=  Göschen Collection . Volume 2132 ). 5th edition. Walter de Gruyter, Berlin / New York 1980, ISBN 3-11-008271-3 , p. 45 ( MR0568703 ).
• Walter Gellert, Herbert Kästner , Siegfried Neuber (Hrsg.): Fachlexikon ABC Mathematik . Verlag Harri Deutsch, Thun, Frankfurt / Main 1978, ISBN 3-87144-336-0 , p. 50 .
• W. Gellert, H. Küstner, M. Hellwich, H. Kästner (eds.): Small encyclopedia of mathematics . Verlag Harri Deutsch, Thun / Frankfurt / Main, ISBN 3-87144-323-9 , p. 232 .
• Siegfried Gottwald , Hans-Joachim Ilgauds , Karl-Heinz Schlote (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch, Thun 1990, ISBN 3-8171-1164-9 , p. 372-373 ( MR1089881 ).
• Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Volume 4: Monge-Ampere Equation - Rings and Algebras. An updated and annotated translation of the Soviet 'Mathematical Encyclopaedia'. Kluwer Academic Publishers , Dordrecht, Boston, London 1995, ISBN 1-55608-010-7 , pp. 439 .
• K. Pohlke : Ten tables on descriptive geometry. Gaertner-Verlag, Berlin 1876 (Google Books.)
• Fritz Reinhardt, Heinrich Soeder (Ed.): Dtv-Atlas for Mathematics. Boards and texts . 8th edition. tape 1 : Basics of algebra and geometry . Deutscher Taschenbuch Verlag , Munich 1990, ISBN 3-423-03007-0 , p. 177 .
• H. Schwarz : Elementary proof of Pohlke's fundamental theorem of axonometry. In: Journal for pure and applied mathematics . tape 63 , 1864, pp. 309-314 ( MR1579271 ).
• Roland Stark: Descriptive Geometry . Schöningh-Verlag, Paderborn 1978, ISBN 3-506-37443-5 .
• Eduard Stiefel : Textbook of the descriptive geometry (=  textbooks and monographs from the field of the exact sciences . Volume 11 ). 3. Edition. Birkhäuser Verlag, Basel (inter alia) 1971, ISBN 3-7643-0368-9 , p. 137 .

References and comments

1. ↑ Heinrich Brauner writes in his textbook on constructive geometry in a footnote (page 51) that Pohlke did publish the fundamental theorem in 1860, albeit without proof.
2. ↑ ^{a b} N. M. Beskin: mapping method . In: PS Alexandroff et al .: Encyclopedia of Elementary Mathematics. Volume IV. 1969, p. 252
3. ↑ ^{a b} Michiel Hazewinkel: Encyclopaedia of Mathematics. vol. 4. 1995, p. 439
4. ^ The Encyclopaedia of Mathematics (p. 439) formulates (PS ^' ) with "Any complete plane quadrilateral ...". In any case, what is meant are complete quadrilaterals in one plane of space. The Encyclopaedia of Mathematics explicitly mentions the essay in Volume IV of the Encyclopedia of Elementary Mathematics as a source .