Summary

Description01-Neuneck-Seite-gegeben Animation.gif	Deutsch: Neuneck (Nonagon), bei gegebener Seitenlänge, Animation der Näherungskonstruktion English: Nonagon, at a given side length, animation of the approximate construction
Date	23 August 2016
Source	Own work
Author	Petrus3743
Other versions

Konstruktionsprinzip

Näherungskonstruktion

Den Winkel 60° mit vier Winkelhalbierende (w₁ ...w₄) verkleinern und den Kreisbogen MON zwischen Winkelhabierende w₃ und w₄ mit folgender Näherungslösung dritteln.

Gerade Hilfslinie g visiert über Punkt O den Punkt N an (quasi ein Lineal an die Punkte O und N angelegt), zwischen O und N daher keine Hilfslinie

${\displaystyle {\overline {PQ}}={\frac {1}{3}}{\overline {OP}}}$

Radius ${\displaystyle {\overline {QR}}={\overline {OP}}}$

Besonderheit

• Es ist eine Konstruktion bei gegebenem Umkreis oder bei gegebener Seitenlänge darstellbar.

Fehler

Bei einem Umkreisradius r = 1 [LE]:

• Konstruierte Neuneckseite s₁ = 0,684040286737346... [LE]
• Neuneckseite s = 2 • sin(20°) = 0,684040286651337... [LE]
• Absoluter Fehler = s₁ − s = 8,6...E-11 [LE]
  • Bei einem Umkreisradius r = 100 000 km wäre der absolute Fehler der 1.Seite ≈ 8,6 mm

Construction principle

Proximity construction

Reduce the angle 60° with four angle bisectors (w₁ ... w₄) and thirds the arc between MON bisector w₃ and w₄ with the following approximate solution.

Straight auxiliary line g aims over the point O to the point N (virtually a ruler at the points O and N applied), between O and N, therefore no auxiliary line.

${\displaystyle {\overline {PQ}}={\frac {1}{3}}{\overline {OP}}}$

Radius ${\displaystyle {\overline {QR}}={\overline {OP}}}$

Special feature

• It is a construction at a given circumscribed circle or at a given side length displayed.

Error

At a circumscribed circle radius r = 1 [unit of length]:

• Constructed side of nonagon s₁ = 0.684040286737346 ... [unit of length]
• Side of nonagon s = 2 • sin(20 °) = 0.684040286651337 ... [unit of length]
• Absolute error = s₁ - s = 8,6 ... E-11 [unit of length]
  • At a circumscribed circle radius r = 100 000 km would be the absolute error of the 1^st side ≈ 8,6 mm

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

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.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue