Kreise 2012 13: Unterschied zwischen den Versionen

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=Aufgaben zum Einstieg=
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==Aufgabe 1==
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Lassen Sie die folgenden Punktmengen in der obigen Geogebraapplikation grafisch darstellen. Um was für geometrische Objekte handelt es ich in jedem Fall? Begründen Sie Ihre Antwort.
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# <math>\left\{P\left(x_P|y_P\right)|x_P^2+y_P^2=1, x_P,x_P \in \mathbb{R} \right\}</math>
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# <math>\left\{P\left(x_P|y_P\right)|\left(x_P-3\right)^2+\left(y_P-2\right)^2=5^2, x_P,y_P \in \mathbb{R}\right\}</math>
  
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==Aufgabe 2==
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Lassen Sie die folgenden Kreise mittels Geoegebra grafisch darstellen, indem Sie jeweils eine entsprechende Gleichung in die Eingabezeile eintragen.
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# Mittelpunkt: <math>M(0|0)</math> Radius: <math>r=5</math>
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# Mittelpunkt: <math>A(2|2)</math> Radius: <math>r=4</math>
  
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=Kreise in der synthetischen Geometrie=
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==Vereinbarung==
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Alle unsere folgenden Betrachtungen beziehen sich auf die Geometrie in der Ebene.
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==Kreisdefinition==
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{{Definition|1=Es seien <math>M</math> ein Punkt und <math>r</math> eine positive reelle Zahl. Unter dem Kreis mit dem Mittelpunkt <math>M</math> und dem radius <math>r</math> versteht man ...}}
  
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=Abstände von Punkten=
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<br /><br />
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Wir beziehen uns auf die obigen Punkte <math>A\left(x_A,y_A\right)</math> und <math>B\left(x_B,y_B\right)</math>.<br />
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Der Punkt <math>C </math> ist der Schnittpunkt der Senkrechten durch <math>A</math> auf die x-Achse mit der Senkrechten von <math>B</math> auf die y-Achse.<br /> Der Punkt <math>C</math> hat damit die Koordinaten <math>\left(x_A-x_B|y_A-y_B\right)</math>.
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<br />
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Das Dreieck <math>\overline{ABC}</math> ist rechtwinklig. Nach dem Satz des Pythagoras gilt:<br />
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(I) <math>|AB|^2=|AC|^2+|BC|^2</math><br />
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Unter Berücksichtigung der Koordinaten von <math>A, B, C</math> schreibt sich (I) wie folgt:<br />
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(II) <math>|AB|^2=\left(x_A-x_B\right)^2+\left(y_A-y_B\right)^2</math><br /><br />
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Abstand zweier Punkte:
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{| class="wikitable"
  
  
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| <math>|AB|=\sqrt{\left(x_A-x_B\right)^2+\left(y_A-y_B\right)^2}</math>
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|-
  
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|}
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=Analytische Beschreibung von Kreisen mittels des Satzes von Pythagoras=
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==Kreis in Mittelpunktslage==
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<br /><br />
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Kreis <math>k</math> in Mittelpunktslage, Radius:r<br /><br />
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{| class="wikitable"
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| <math>k=\left{A\left(x_A|y_A\right)|x_A^2+y_A^2=r^2\right}</math>
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|-
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|}
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==allgemeine Lage==
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=DerAlgorithmus von Bresenham=
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" 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[[Kategorie:Linalg]]
 
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</div>

Aktuelle Version vom 1. November 2012, 08:19 Uhr

Inhaltsverzeichnis

Aufgaben zum Einstieg

Aufgabe 1

Lassen Sie die folgenden Punktmengen in der obigen Geogebraapplikation grafisch darstellen. Um was für geometrische Objekte handelt es ich in jedem Fall? Begründen Sie Ihre Antwort.

  1. \left\{P\left(x_P|y_P\right)|x_P^2+y_P^2=1, x_P,x_P \in \mathbb{R} \right\}
  2. \left\{P\left(x_P|y_P\right)|\left(x_P-3\right)^2+\left(y_P-2\right)^2=5^2, x_P,y_P \in \mathbb{R}\right\}

Aufgabe 2

Lassen Sie die folgenden Kreise mittels Geoegebra grafisch darstellen, indem Sie jeweils eine entsprechende Gleichung in die Eingabezeile eintragen.

  1. Mittelpunkt: M(0|0) Radius: r=5
  2. Mittelpunkt: A(2|2) Radius: r=4

Kreise in der synthetischen Geometrie

Vereinbarung

Alle unsere folgenden Betrachtungen beziehen sich auf die Geometrie in der Ebene.

Kreisdefinition

Definition

Es seien M ein Punkt und r eine positive reelle Zahl. Unter dem Kreis mit dem Mittelpunkt M und dem radius r versteht man ...

Abstände von Punkten



Wir beziehen uns auf die obigen Punkte A\left(x_A,y_A\right) und B\left(x_B,y_B\right).
Der Punkt C ist der Schnittpunkt der Senkrechten durch A auf die x-Achse mit der Senkrechten von B auf die y-Achse.
Der Punkt C hat damit die Koordinaten \left(x_A-x_B|y_A-y_B\right).
Das Dreieck \overline{ABC} ist rechtwinklig. Nach dem Satz des Pythagoras gilt:
(I) |AB|^2=|AC|^2+|BC|^2
Unter Berücksichtigung der Koordinaten von A, B, C schreibt sich (I) wie folgt:
(II) |AB|^2=\left(x_A-x_B\right)^2+\left(y_A-y_B\right)^2

Abstand zweier Punkte:

|AB|=\sqrt{\left(x_A-x_B\right)^2+\left(y_A-y_B\right)^2}


Analytische Beschreibung von Kreisen mittels des Satzes von Pythagoras

Kreis in Mittelpunktslage



Kreis k in Mittelpunktslage, Radius:r

Fehler beim Parsen(Syntaxfehler): k=\left{A\left(x_A|y_A\right)|x_A^2+y_A^2=r^2\right}

allgemeine Lage

DerAlgorithmus von Bresenham

Von „http://geometrie.zum.de/index.php?title=Kreise_2012_13&oldid=18132“