Run only the animation
In[]:=
Development:
Parametric equations of the geometric locus of G point, considering the ellipse equation
2
x
2
a
+
2
y
2
b
=1
.Let
CD
= k, then:
In[]:=
D=={c+kCos[φ],kSin[φ]};​​Solve
2
(c+kCos[φ])
2
a
+
2
(kSin[φ])
2
b
==1,k//FullSimplify
Solve
:Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.
Out[]=
k
-0.5Cos[φ]-0.204124
37.-13.Cos[2.φ]
1.88889-0.888889Cos[2φ]
,k
-0.5Cos[φ]+0.204124
37.-13.Cos[2.φ]
1.88889-0.888889Cos[2φ]

In[]:=
k-
b2bcCos[φ]+
2
a
2
a
+
2
b
-
2
c
+(-
2
a
+
2
b
+
2
c
)Cos[2φ]

2(
2
b
2
Cos[φ]
+
2
a
2
Sin[φ]
)
,k
b-2bcCos[φ]+
2
a
2
a
+
2
b
-
2
c
+(-
2
a
+
2
b
+
2
c
)Cos[2φ]

2(
2
b
2
Cos[φ]
+
2
a
2
Sin[φ]
)
;
In[]:=
k[φ_]:=
b-2bcCos[φ]+
2
a
2
a
+
2
b
-
2
c
+(-
2
a
+
2
b
+
2
c
)Cos[2φ]

2(
2
b
2
Cos[φ]
+
2
a
2
Sin[φ]
)
;
Let
CE
= l, then:
In[]:=
E=={c+lCos[φ+β],lSin[φ+β]};​​Solve
2
(c+lCos[φ+β])
2
a
+
2
(lSin[φ+β])
2
b
==1,l//FullSimplify
Solve
:Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.
Out[]=
l
-0.5Sin[0.523599-1.φ]-0.204124
37.+13.Sin[0.523599+2.φ]
1.88889+0.888889Cos[1.0472-2.φ]
,l
-0.5Sin[0.523599-1.φ]+0.204124
37.+13.Sin[0.523599+2.φ]
1.88889+0.888889Cos[1.0472-2.φ]

In[]:=
l-
2
b
cCos[β+φ]+
2
a
2
b
2
Cos[β+φ]
+(a-c)(a+c)
2
Sin[β+φ]
2
a
2
b
2
b
2
Cos[β+φ]
+
2
a
2
Sin[β+φ]
,l
2
b
-cCos[β+φ]+
2
a
2
b
2
Cos[β+φ]
+(a-c)(a+c)
2
Sin[β+φ]
2
a
2
b
2
b
2
Cos[β+φ]
+
2
a
2
Sin[β+φ]
;
In[]:=
l[φ_]:=
2
b
-cCos[β+φ]+
2
a
2
b
2
Cos[β+φ]
+(a-c)(a+c)
2
Sin[β+φ]
2
a
2
b
2
b
2
Cos[β+φ]
+
2
a
2
Sin[β+φ]
;
In both cases k and l, the positive roots were chosen (because they are magnitudes).
Animation:
In[]:=
Manipulatea=1;b=0.6;c=0.5;β=
π
3
;C1={c,0};A={a,0};B={0,b};O1={0,0};​​ζ[θ_]:={aCos[θ],bSin[θ]};​​k[φ_]:=
b-2bcCos[φ]+
2
a
2
a
+
2
b
-
2
c
+(-
2
a
+
2
b
+
2
c
)Cos[2φ]

2(
2
b
2
Cos[φ]
+
2
a
2
Sin[φ]
)
;​​l[φ_]:=
2
b
-cCos[β+φ]+
2
a
2
b
2
Cos[β+φ]
+(a-c)(a+c)
2
Sin[β+φ]
2
a
2
b
2
b
2
Cos[β+φ]
+
2
a
2
Sin[β+φ]
;​​D1[φ_]:={c+k[φ]Cos[φ],k[φ]Sin[φ]};​​E1[φ_]:={c+l[φ]Cos[φ+β],l[φ]Sin[φ+β]};​​G[φ_]:=D1[φ]+E1[φ]-C1;​​CD[φ_]:=D1[φ]-C1;​​CE[φ_]:=E1[φ]-C1;CG[φ_]:=G[φ]-C1;​​δ=ResourceFunction["PlotVector"][{CD[φ],CE[φ],CG[φ]},C1];​​Show[{​​ParametricPlot[ζ[θ],{θ,0,2π},PlotStyle{Blue,Thick}],​​ParametricPlot[G[θ],{θ,0,φ},ColorFunction->"BlueGreenYellow"],​​δ,​​Graphics[{Red,PointSize[0.01],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],​​Point[B],Text[Style["B",14,Bold],B,{0,-1.2}],​​Point[C1],Text[Style["C",14,Bold],C1,{0,-1.2}],​​Point[D1[φ]],Text[Style["D",14,Bold],D1[φ],{0,-1.2}],​​Point[E1[φ]],Text[Style["E",14,Bold],E1[φ],{0,-1.2}],​​Point[G[φ]],Text[Style["G",14,Bold],G[φ],{0,-1.2}],​​{Black,InfiniteLine[{E1[φ],G[φ]}],InfiniteLine[{D1[φ],G[φ]}]}}]},AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-3,2},{-1,2}},ImageSize700],Style["Geometric locus of G point",Bold,Large],{{φ,0.00002,"Value (φ)"},0.000001,2π,0.00001},ControlPlacementTop,SaveDefinitions->True
Out[]=
​
Geometric locus of G point
Value (φ)