Polarkoordinaten

\(\(PolarPlot[1, {t, 0, π\/2}, PlotStyle -> Farbig, 
        PlotRange -> {{0, 1.1}, {0, 1}}, AxesLabel -> {x, y}, 
        Ticks -> None, \ 
Epilog -> {Thickness[0.01], 
            PointSize[0.03], 
Point[{\(1\/2\) \@3, 1\/2}], 
            Text[StyleForm["P(r,φ)", 
                FontFamily -> "Arial", FontSize -> 16, 
                FontWeight -> "Bold"], {0.97, 0.6}, {0, 1}, {1, 
                0}], 


Line[{{0, 0}, {\(1\/2\) \@3, 1\/2}}], 
Text[
              StyleForm["r", FontFamily -> "Arial", 
                FontSize -> 16, FontWeight -> "Bold"], {1\/2, 
                0.4}, {0, 1}, {1, 
                0}], 

RGBColor[0, 0.8, 
              0], 
Line[{{\(1\/2\) \@3, 0}, {\(1\/2\) \@3, 
                  1\/2}}], 

Text[
              StyleForm["r sin φ", 
                FontFamily -> "Arial", FontSize -> 16, 
                FontWeight -> "Bold"], {0.8, 0.29}, {0, 1}, {0, 
                1}], 
RGBColor[0, 0, 
              0.8], 
Line[{{0, 0}, {\(1\/2\) \@3, 
                  0}}], 

Text[
              StyleForm["r cos φ", 
                FontFamily -> "Arial", FontSize -> 16, 
                FontWeight -> "Bold"], {0.6, 0.08}, {0, 1}, {1, 
                0}], 
RGBColor[0.8, 0, 
              0.8], 
Circle[{0, 0}, 
              0.25, {0, π\/6}], 
Text[
              StyleForm["φ", FontFamily -> "Arial", 
                FontSize -> 16, FontWeight -> "Bold"], {0.3, 
                0.13}, {0, 1}, {1, 0}]
}\ ];\)\)

Umwandlung in Polarkoordinaten

regel = {x -> r Cos[φ],  
          y -> r Sin[φ]};

Urspungsgeraden

Sie lassen sich nur durch φ=constant beschreiben, vom CAS werden sie nicht gezeichnet.

Andere Geraden

y == m x  + b /. regel

Solve[%, r]

Ursprungs-Kreis

r = constant

PolarPlot[{1, 2, 3, 4}, {t, 0, π}, 
        PlotStyle -> Farbig];

Verschobener Kreis

vKreis = (((x - mx))^2) + (((y - my))^2) == (R^2);

vKreis /. regel // FullSimplify

Solve[%, r] // FullSimplify

% /. mx -> 0

% /. my -> R

PolarPlot[2 Sin[t], {t, 0, 2  π}, 
        PlotStyle -> Farbig];

Allgemeine Konchoiden

Allgemein  r= strasse[φ] sei die Straße, k die Leinenlänge, Baum im Ursprung.
Dann ist  r=strasse[φ]±k   die Gleichung der Konchoide

Gerade Straße y=a, Baum im Ursprung

hund = (a/Sin)[φ] - k; 
    fiffi = (a/Sin)[φ] + k;

a = 4; Solve[(a/Sin)[φ] == 8, φ]

zurueck[r_, φ_] := {r Cos[φ],  
          r Sin[φ]}; 
zurueck[4, (π/6)]

% // N

zurueck[8, (π/6)]

PolarPlot[{hund /. k -> 4 // Evaluate, 
        (a/Sin)[φ]}, {φ, 0.2, 2.5}, 
      PlotStyle -> 
        Farbig, 
Epilog -> {Line[{{-9, 4}, {6.5, 
                4}}], 
Line[{{4 (Sqrt[3]), 4}, {0, 
                0}}], 
Circle[{0, 0}, 
            2, {0, (π/6)}]
, 
          Thickness[0.02], 
RGBColor[0, 0.8, 
            0], 
PointSize[0.03], 
          Point[{4 (Sqrt[3]), 
              4}], 
Line[{{4 (Sqrt[3]), 4}, {2 (Sqrt[ 3]), 2}}], 
          Point[{2 (Sqrt[3]), 
              2}]



}]

(a/Sin)[(π/6)] - a

a = 4;
PolarPlot[{{hund, fiffi}  /. k -> 6, {hund, fiffi} /. 
              k -> 4, {hund, fiffi}  /. k -> 2} // Flatten // 
        Evaluate, {φ, 0.3, 3}, 
      PlotStyle -> 
        Farbig, 
Epilog -> {Line[{{-9, 4}, {6.5, 
                4}}]}]

Pascalsche Schnecken

{{hund, fiffi}  /. k -> 5, {hund, fiffi} /. 
            k -> 4, {hund, fiffi}  /. k -> 3}} // Flatten

{{hund, fiffi}  /. k -> 5, {hund, fiffi}  /. k -> 5} // 
      Flatten