File:01-Elfeck-wiki.svg
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Summary
[edit]Description01-Elfeck-wiki.svg |
Deutsch: Elfeck, eine Näherungskonstruktion mit Anwendung des dritten Strahlensatzes English: Hendecagon, an approximate construction with application of the third intercept theorem |
Date | |
Source | Own work |
Author | Petrus3743 |
Other versions | |
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Näherungskonstruktion
[edit]- Konstruktionsprinzip: Dritter Strahlensatz und Dezimalbruch
- Die Qualität der Näherung wird durch die Wahl des Dezimalbruchs (Anzahl der Nullen im Nenner) vorherbestimmt.
- Im dargestellten Beispiel ist der Dezimalbruch gewählt, deshalb sind elf Nachkommastellen gleich dem Wert = 0,563465113682859... [LE]
Fehler
[edit]Bezogen auf den Einheitskreis r = 1 [LE]:
- Konstruierte Seite des Elfecks sk = 0,56346511368 [LE]
- Seite des Elfecks s = = 0,563465113682859... [LE]
- Absoluter Fehler der konstruierten Seite = sk - s = -0,00000000000286... = -2,86...E-12 [LE]
Beispiel zur Verdeutlichung: Bei einem Umkreisradius r = 1 Mio. km wäre der Fehler der Seite sk ≈ -2,9 mm
Berechnung
[edit]Die Berechnung der konstruierten Seite des Elfecks sk geschieht, aufgrund des Konstruktionsprinzips, schrittweise durch die geometrischen Additionen / Subtraktionen der einzelnen Zwischenergebnissen auf den Zahlenstrahlen s1 bzw. s4.
Besonderheit
[edit]Mit geringer Änderung der Arbeitsschritte ist auch eine Näherungskonstruktion bei gegebener Seite des Elfecks machbar:
- Die Basiskonstruktion ohne Umkreis und ohne den Mittelachsen verwenden.
- Den mittigen Hilfsstrahl mit den beiden Scheitelpunkten und ergänzen.
- Den Zähler als Strecke auf die Strecke konstruieren.
- Die Parallele zur Strecke einzeichnen, es ergibt sich die Strecke .
- Die Strecke ist der Umkreisradius .
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Proximity construction
[edit]- Construction principle: Intercept theorem, point 4. "If you have more than two lines ..." and decimal fraction.
- The quality of the approximation is determined in advance through the choice of decimal fraction (number of zeros in the denominator).
- In the represented example the decimal fraction is selected, so are eleven decimal places equal to the value = 0.563465113682859... [unit of length]
Error
[edit]Based on the unit circle r = 1 [unit of length]:
- Constructed side of the hendecagon sk = 0.56346511368 [unit of length]
- Side of the hendecagon s = = 0.563465113682859... [unit of length]
- Absolute error of the constructed side = sk - s = -0.00000000000286... = -2.86...E-12 [unit of length]
Example to illustrating: At a circumscribed circle radius r = 1 billion km, would be the error of the side sk ≈ -2.9 mm
Calculation
[edit]The calculation of the constructed side of the hendecagon sk takes place, due to the construction principle, step by step through the geometric additions / subtractions of the individual intermediate results, on the number line s1 respectively s4.
Special feature
[edit]With little change in construction steps also an approximate construction at a given side of the hendecagon is feasible:
- Use the basic construction without circumscribed circle and without the central axes.
- The auxiliary beam at the center add to with the two vertices points and .
- The counter as distance construct on distance .
- The parallel to the distance , the outcome of this is the distance .
- The distance is the circumscribed circle radius .
Licensing
[edit]- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 15:46, 26 June 2015 | 1,663 × 1,405 (285 KB) | Petrus3743 (talk | contribs) | Grafikfehler korrigiert | |
15:34, 26 June 2015 | 1,663 × 1,405 (285 KB) | Petrus3743 (talk | contribs) | Konstruktion überarbeitet | ||
00:41, 13 January 2015 | 1,082 × 906 (253 KB) | Petrus3743 (talk | contribs) | Zwei Funktionskreise ergänzt, Bildgröße 1 Einheit = 8 cm | ||
12:40, 27 December 2014 | 1,070 × 953 (246 KB) | Petrus3743 (talk | contribs) | ohne Transparent | ||
12:26, 27 December 2014 | 1,067 × 953 (246 KB) | Petrus3743 (talk | contribs) | Bezeichnung für s5 ergänzt, Bildgröße: 1 Einheit = 8 cm | ||
10:45, 27 December 2014 | 1,069 × 954 (246 KB) | Petrus3743 (talk | contribs) | User created page with UploadWizard |
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Width | 46.940092451809214cm |
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Height | 39.65628500239054cm |