Conchoids::usage="
Reference:WolframMathWorld:Conchoid;​ConchoidofdeSluzePersonal.;";
Example 1: Sine’s conchoid
In[]:=
SwatchLegend[{Green,Red,Black},{"Positive Conchoid","Negative Conchoid","Original function"}]​​Manipulatef[t_]:=t;g[t_]:=Sin[t];
x
0
=3;
y
0
=-3;k=2;
ζ
+
[t_]:=f[t]+
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]+
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
;​​
ζ
-
[t_]:=f[t]-
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]-
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
;​​Show{ParametricPlot[
ζ
+
[t],{t,0,t1},PlotStyle{Green,Thick},PerformanceGoal->"Quality"],ParametricPlot[
ζ
-
[t],{t,0,t1},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],ParametricPlot[{f[t],g[t]},{t,0,2π},PlotStyle{Black,Thick}],Graphics[{Green,PointSize[0.015],Point[
ζ
+
[t1]]}],Graphics[{Red,PointSize[0.015],Point[
ζ
-
[t1]]}],Graphics[{Black,Thick,Line[{
ζ
-
[t1],{
x
0
,
y
0
},
ζ
+
[t1]}]}],Graphics[{Black,PointSize[0.015],Point[{
x
0
,
y
0
}]}],Graphics[{Black,PointSize[0.015],Point[{f[t1],g[t1]}]}]},AxesTrue,AxesOrigin{0,0},AspectRatioAutomatic,ImageSize700,PlotRange-
π
3
,2.5π,{-4,4},PlotLabelStyle[Framed["Conchoid"],16,Blue,BackgroundYellow],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacement->Top​​
Out[]=
Positive Conchoid
Negative Conchoid
Original function
Out[]=
​
Value (t1)
Example 2: Circle’s conchoid
In[]:=
SwatchLegend[{Green,Red,Black},{"Positive Conchoid","Negative Conchoid","Original function"}]​​Manipulatef[t_]:=Cos[t]+1.25;g[t_]:=Sin[t];
x
0
=0;
y
0
=0;k=1.5;
ζ
+
[t_]:=f[t]+
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]+
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
;​​
ζ
-
[t_]:=f[t]-
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]-
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
;​​Show[{ParametricPlot[
ζ
+
[t],{t,-2π,t1},PlotStyle{Green,Thick},PerformanceGoal->"Quality"],ParametricPlot[
ζ
-
[t],{t,-2π,t1},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],ParametricPlot[{f[t],g[t]},{t,0,2π},PlotStyle{Black,Thick}],Graphics[{Green,PointSize[0.015],Point[
ζ
+
[t1]]}],Graphics[{Red,PointSize[0.015],Point[
ζ
-
[t1]]}],Graphics[{Black,Thick,Line[{
ζ
-
[t1],{
x
0
,
y
0
},
ζ
+
[t1]}]}],Graphics[{Black,PointSize[0.015],Point[{
x
0
,
y
0
}]}],Graphics[{Black,PointSize[0.015],Point[{f[t1],g[t1]}]}]},AxesTrue,AxesOrigin{0,0},AspectRatioAutomatic,ImageSize700,PlotRange{{-2π,2π},{-4,4}},PlotLabelStyle[Framed["Conchoid"],16,Blue,BackgroundYellow]],{{t1,-2π+0.00002,"Value (t1)"},-2π+0.00001,2π,0.00001},ControlPlacement->Top​​
Out[]=
Positive Conchoid
Negative Conchoid
Original function
Out[]=
​
Value (t1)
Example 3: Ellipse’s conchoid
In[]:=
SwatchLegend[{Green,Red,Black},{"Positive Conchoid","Negative Conchoid","Original function"}]​​Manipulatef[t_]:=3Cos[t];g[t_]:=2Sin[t];
x
0
=0;
y
0
=1.5;k=1.5;
ζ
+
[t_]:=f[t]+
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]+
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
;​​
ζ
-
[t_]:=f[t]-
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]-
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
;​​Show[{ParametricPlot[
ζ
+
[t],{t,0,t1},PlotStyle{Green,Thick},PerformanceGoal->"Quality"],ParametricPlot[
ζ
-
[t],{t,0,t1},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],ParametricPlot[{f[t],g[t]},{t,0,2π},PlotStyle{Black,Thick}],Graphics[{Green,PointSize[0.015],Point[
ζ
+
[t1]]}],Graphics[{Red,PointSize[0.015],Point[
ζ
-
[t1]]}],Graphics[{Black,Thick,Line[{
ζ
-
[t1],{
x
0
,
y
0
},
ζ
+
[t1]}]}],Graphics[{Black,PointSize[0.015],Point[{
x
0
,
y
0
}]}],Graphics[{Black,PointSize[0.015],Point[{f[t1],g[t1]}]}]},AxesTrue,AxesOrigin{0,0},AspectRatioAutomatic,ImageSize700,PlotRange{{-2π,2π},{-4,5}},PlotLabelStyle[Framed["Conchoid"],16,Blue,BackgroundYellow]],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacement->Top​​
Example 4: Nicomedes conchoid