File:01-Neuneck.svg

Summary[edit]

Description01-Neuneck.svg	Deutsch: Neuneck (Nonagon) bei gegebenem Umkreis, Näherungskonstruktion English: Nonagon at a given circumscribed circle, approximate construction
Date	22 October 2015
Source	Own work
Author	Petrus3743
Other versions
SVG development InfoField	The SVG code is valid.   This trigonometry was created with GeoGebra by Petrus3743.   This SVG trigonometry uses the path text method.

Konstruktionsprinzip[edit]

Näherungskonstruktion[edit]

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[edit]

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

Fehler[edit]

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[edit]

Proximity construction[edit]

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[edit]

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

Error[edit]

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[edit]

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