Summary edit

Description01-257-Eck.svg	Deutsch: Das 257-Eck, Näherungskonstruktion der ersten Seite English: The 257-gon, approximate construction of the first side
Date	3 January 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.

Ergebnis, Score edit

Bezogen auf den Einheitskreis R = 1 [LE] edit

• Konstruierte Seite des 257-Ecks ${\displaystyle {\overline {AE_{1}}}=a=2{,}44475843350370...E-2\;[LE]}$
• Seite des 257-Ecks ${\displaystyle a_{SOLL}=2\cdot \sin \left({\frac {180^{\circ }}{257}}\right)=2{,}44475829850863...E-2\;[LE]}$
• Absoluter Fehler der konstruierten Seite ${\displaystyle F_{a}=a-a_{SOLL}=1{,}3499...E-9\;[LE]}$
• Konstruierter Zentriwinkel ${\displaystyle \alpha =1{,}40077828746899...^{\circ }}$
• Zentriwinkel des 257-Ecks ${\displaystyle \alpha _{SOLL}={\frac {360^{\circ }}{257}}=1{,}40077821011673...^{\circ }}$
• Absoluter Winkelfehler vom konstruierten Zentriwinkel ${\displaystyle F_{\alpha }=\alpha -\alpha _{SOLL}=7{,}735...E-8^{\circ }}$

(1 Winkelsekunde = ${\displaystyle {\frac {1^{\circ }}{3600}}}$ = 0,000277...° = 2,77...E-4°)

Beispiel zur Verdeutlichung des Fehlers edit

Bei einem Umkreisradius R = 1000 km wären die absoluten Fehler von:

• 1. Seite ≈ 1,35 mm
• 257. Seite ≈ -350 mm

Based on the unit circle R = 1 [unit of length] edit

• Constructed side of 257-gon ${\displaystyle {\overline {AE_{1}}}=a=2.44475843350370...E-2...\;[unit\;of\;length]}$
• Side of the 257-gon ${\displaystyle a_{SHOULD}=2\cdot \sin \left({\frac {180^{\circ }}{257}}\right)=2.44475829850863...E-2\;[unit\;of\;length]}$
• Absolute error of the constructed side ${\displaystyle F_{a}=a-a_{SOLL}=1.349...E-9\;[unit\;of\;length]}$
• Constructed central angle ${\displaystyle \alpha =1.40077828746899...^{\circ }}$
• Central angle of the 257-gon ${\displaystyle \alpha _{SHOULD}={\frac {360^{\circ }}{257}}=1.40077821011673...^{\circ }}$
• Absolute angular error of the constructed central angle ${\displaystyle F_{\alpha }=\alpha -\alpha _{SOLL}=7.735...E-8^{\circ }}$

(1 arcsecond = ${\displaystyle {\frac {1^{\circ }}{3600}}}$ = 0.000277...° = 2.77...E-4°)

Example to illustrate the error edit

At a circumscribed circle R = 1000 km, would be the absolute errors of:

• Side 1st ≈ 1.35 mm
• Side 257th ≈ -350 mm

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