File:01-Elfeck-Animation.gif
01-Elfeck-Animation.gif (665 × 555 pixels, file size: 56 KB, MIME type: image/gif, looped, 98 frames, 2 min 34 s)
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Contents
Summary
editDescription01-Elfeck-Animation.gif |
Deutsch: Elfeck, Animation einer Näherungskonstruktion
English: Hendekagon, Animation of an approximate construction |
Date | |
Source | Own work |
Author | Petrus3743 |
Other versions |
![]() Hendecagon, an approximate construction |
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
editBezogen 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
editDie 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
editMit geringer Änderung der Arbeitsschritte ist auch eine Näherungskonstruktion mit 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
.
---
Proximity construction
edit- Construction principle: third intercept theorem 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
editBased 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 million km, would be the error of the side sk ≈ -2.9 mm
Calculation
editThe 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
editWith little change in construction steps also an approximate construction with 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![w:en:Creative Commons](https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/CC_some_rights_reserved.svg/90px-CC_some_rights_reserved.svg.png)
![attribution](https://upload.wikimedia.org/wikipedia/commons/thumb/1/11/Cc-by_new_white.svg/24px-Cc-by_new_white.svg.png)
![share alike](https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Cc-sa_white.svg/24px-Cc-sa_white.svg.png)
- 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 | |
---|---|---|---|---|---|
current | 15:24, 26 June 2015 | ![]() | 665 × 555 (56 KB) | Petrus3743 (talk | contribs) | Pausenlängen angepasst |
14:22, 26 June 2015 | ![]() | 665 × 555 (56 KB) | Petrus3743 (talk | contribs) | 72 dpi, 1 Einheit = 5 cm, 99 Bilder | |
22:43, 25 June 2015 | ![]() | 1,201 × 1,040 (115 KB) | Petrus3743 (talk | contribs) | User created page with UploadWizard |
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