Summary

Description01-Neuneck-Animation.gif	Deutsch: Neuneck (Nonagon) bei gegebenem Umkreis, Animation der Näherungskonstruktion English: Nonagon at a given circumscribed circle, animation of the approximate construction
Date	14 May 2015
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

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