词条 | Pohlke's theorem |
释义 |
Pohlke's theorem is the fundamental theorem of axonometry. It was established 1853 by the German painter and teacher of descriptive geometry Karl Wilhelm Pohlke. The first proof of the theorem was published 1864 by the German mathematician Hermann Amandus Schwarz, who was a student of Pohlke. Therefore the theorem is sometimes called theorem of Pohlke and Schwarz, too. The theorem
For a mapping of a unit cube, one has to apply an additional scaling either in the space or in the plane. Because a parallprojection and a scaling preserves ratios one can map an arbitrary point by the axonometric procedure below. Pohlke's theorem can be stated in terms of linear algebra as:
Application to axonometryPohlke's theorem is the justification for the following easy procedure to construct a scaled parallel projection of a 3-dimensional object using coordinates,:[2][3]
go in -direction, then go in -direction, then go in -direction and 4. mark the point as . In order to get undistorted pictures, one has to choose the images of the axes and the forshortenings carefully (see Axonometry). In order to get an orthographic projection only the images of the axes are free and the forshortenings are determined. (see de:orthogonale Axonometrie). Remarks on Schwarz's proofSchwarz formulated and proved the more general statement:
and used a theorem of L’Huilier:
Notes1. ^G. Pickert: Vom Satz von Pohlke zur linearen Algebra, Didaktik der Mathematik 11 (1983), 4, pp. 297–306. 2. ^Ulrich Graf, Martin Barner: Darstellende Geometrie. Quelle & Meyer, Heidelberg 1961, {{ISBN|3-494-00488-9}}, p.144. 3. ^Roland Stärk: Darstellende Geometrie, Schöningh, 1978, {{ISBN|3-506-37443-5}}, p.156. 4. ^{{cite journal |url=http://math.unipa.it/~grim/quad17_sklenarikova-pemova_07.pdf |first1=Zita |last1=Sklenáriková |first2=Marta |last2=Pémová |title=The Pohlke–Schwarz Theorem and its Relevancy in the Didactics of Mathematics|journal=Quaderni di Ricerca in Didattica|publisher=G.R.I.M. (Department of Mathematics, University of Palermo, Italy)|year=2007|number=17|page=155}} References
External links
2 : Graphical projections|Linear algebra |
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