File:01 Pi-Approximation.svg

Description01 Pi-Approximation.svg	Deutsch: Näherungskonstruktion der halben Kreiszahl (${\displaystyle {\tfrac {\pi }{2}}}$) mit Zirkel und Lineal English: Proximity construction of the half circle number (${\displaystyle {\tfrac {\pi }{2}}}$) with compass and ruler
Date	1 March 2022
Source	Own work
Author	Petrus3743
SVG development InfoField	The SVG code is valid.   This trigonometry was created with GeoGebra by Petrus3743.   This SVG trigonometry uses the path text method.

1. Einheitskreis um ${\displaystyle M}$ mit Radius ${\displaystyle {\overline {AM}}=1}$.
2. Halbgerade durch ${\displaystyle A}$ und ${\displaystyle M}$ ergibt Schnittpunkt ${\displaystyle B}$.
3. Halbgerade senkrecht zu ${\displaystyle {\overline {AB}}}$ durch ${\displaystyle M}$ ergibt Schnittpunkte ${\displaystyle C}$ und ${\displaystyle D}$.
4. Strecken ${\displaystyle {\overline {AE}}={\frac {1}{10}}\cdot {\overline {AM}}={\overline {AF}}={\overline {FG}}={\overline {GH}}={\overline {HI}}}$, Kreis um ${\displaystyle M}$ durch ${\displaystyle E}$.
5. Strecken ${\displaystyle {\overline {EJ}}={\overline {JA}}={\overline {IK}}={\overline {HL}}}$, Kreis um ${\displaystyle M}$ durch ${\displaystyle J}$.
6. Bestimmen der Funktionspunkte:

Es beginnt mit Punkt ${\displaystyle N}$, dessen Abstand zu Punkt ${\displaystyle B}$ ist gleich der Strecke ${\displaystyle {\overline {DI}}}$. In der Darstellung beschrieben als ${\displaystyle |BN|={\overline {DI}}}$. Auf diese Art und Weise werden auch die weiteren Funktionspunkte von ${\displaystyle O}$ als ${\displaystyle |DO|={\overline {MG}}}$ bis ${\displaystyle Z}$ als ${\displaystyle |DZ|={\overline {EH}}}$ (Reihenfolge siehe Kurzbeschreibung in der Darstellung) festgelegt.

7. Einzeichnen der Kreissekanten:

Es beginnt mit der Sekante ab ${\displaystyle Z}$ durch ${\displaystyle Y}$ bis sie die äußere Kreislinie in ${\displaystyle A_{1}}$ schneidet. Die nächste Sekante läuft ab dem zuletzt erhaltenen Schnittpunkt ${\displaystyle A_{1}}$ durch ${\displaystyle X}$ bis sie wieder die äußere Kreislinie in ${\displaystyle B_{1}}$ schneidet. Auf diese Art und Weise werden auch die Punkte von ${\displaystyle C_{1}}$ bis ${\displaystyle L_{1}}$ (Reihenfolge ist anhand des Verlaufs der Sekanten zu entnehmen) bestimmt.

8. Letzte Sekante von ${\displaystyle L_{1}}$ bis ${\displaystyle E}$ schneidet die Mittelachse des Einheitskreises in ${\displaystyle M_{1}}$.
9. Der abschließende Halbkreis um ${\displaystyle M}$ ab ${\displaystyle M_{1}}$ liefert die Strecke ${\displaystyle {\overline {M_{1}N_{1}}}}$, deren Länge nahezu der halben Kreiszahl ${\displaystyle {\frac {\pi }{2}}}$ entspricht .

• In GeoGebra konstruierte Strecke ${\displaystyle {\overline {M_{1}N_{1}}}=1{,}570796326794897\approx {\frac {\pi }{2}},\;}$ (Anzeige max. 15 Nachkommastellen), letzte Stelle gerundet.
• Wert von ${\displaystyle {\frac {\pi }{2}}=1{,}5707963267948966\ldots }$
• Der absolute Fehler der konstruierten halben Kreiszahl ${\displaystyle \approx {\frac {\pi }{2}}}$ ist in GeoGebra aufgrund der Anzeigebegrenzung und der Rundung nicht verifizierbar.

Bei einem Umkreisradius r = 1 Mrd. km (das Licht bräuchte für diese Strecke ca. 56 min), wäre der absolute Fehler der konstruierten Länge ${\displaystyle {\overline {M_{1}N_{1}}}}$ < 1 mm.

1. Unit circle around ${\displaystyle M}$ with radius ${\displaystyle {\overline {AM}}=1}$.
2. Half-line through ${\displaystyle A}$ and ${\displaystyle M}$ results in intersection ${\displaystyle B}$.
3. Half-line perpendicular to ${\displaystyle {\overline {AB}}}$ through ${\displaystyle M}$ gives intersection ${\displaystyle C}$ and ${\displaystyle D}$.
4. Line segments ${\displaystyle {\overline {AE}}={\frac {1}{10}}\cdot {\overline {AM}}={\overline {AF}}={\overline {FG}}={\overline {GH}}={\overline {HI}}}$, circle around ${\displaystyle M}$ by ${\displaystyle E}$.
5. Line segments ${\displaystyle {\overline {EJ}}={\overline {JA}}={\overline {IK}}={\overline {HL}}}$, circle around ${\displaystyle M}$ by ${\displaystyle J}$.
6. Determine the function points:

It starts with point ${\displaystyle N}$, its distance to point ${\displaystyle B}$ is equal to line segment ${\displaystyle {\overline {DI}}}$. In the representation described as ${\displaystyle |BN|={\overline {DI}}}$. In this way also the further function points are defined, ${\displaystyle O}$ as ${\displaystyle |DO|={\overline {MG}}}$ up to the last ${\displaystyle Z}$ as ${\displaystyle |DZ|={\overline {EH}}}$ (order see short description in the representation).

7. Drawing the circle secants:

It starts with the secant from ${\displaystyle Z}$ through ${\displaystyle Y}$ until it intersects the outer circle line in ${\displaystyle A_{1}}$. The next secant runs from the last obtained intersection point ${\displaystyle A_{1}}$ through ${\displaystyle X}$ until it again intersects the outer circle line in ${\displaystyle B_{1}}$. In this way, the points from ${\displaystyle C_{1}}$ to ${\displaystyle L_{1}}$ (order can be inferred from the course of the secants) are also determined.

8. Last secant from ${\displaystyle L_{1}}$ to ${\displaystyle E}$ intersects the center axis of the unit circle in ${\displaystyle M_{1}}$.
9. The final semicircle around ${\displaystyle M}$ starting at ${\displaystyle M_{1}}$ yields the line segment ${\displaystyle {\overline {M_{1}N_{1}}}}$ whose length is nearly half the circle number ${\displaystyle {\frac {\pi }{2}}}$ .

• Line segment constructed in GeoGebra ${\displaystyle {\overline {M_{1}N_{1}}}=1.570796326794897\approx {\frac {\pi }{2}},\;}$ (display max 15 decimal places), last digit rounded.
• Value of ${\displaystyle {\frac {\pi }{2}}=1.5707963267948966\ldots }$.
• The absolute error of the constructed half circle number ${\displaystyle \approx {\frac {\pi }{2}}}$ is not verifiable in GeoGebra due to display limitations and rounding.

For a radius of revolution r = 1 billion km (the light would need about 56 min for this line segment), the absolute error of the constructed length would be ${\displaystyle {\overline {M_{1}N_{1}}}}$ < 1 mm.

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