GeometricLocus::usage="The geometric locus of the midpoint of a focal radius, let's say , as it rotates around F'. References: Wexler, Charles. Geometría Analítica. Un enfoque vectorial. Montaner y Simón, S.A. Barcelona: p.154, 1977; Personal. ";
FP
Development:
Suppose it’s the ellipse:
2
x
2
a
2
y
2
b
Method 1:
In parametric equations:
ζ[t_]:={aCos[t],bSin[t]};
A point on the ellipse is:
P={aCos[t],bSin[t]};
A focus of the ellipse is:
F'=(c,0);
The midpoint of focal radius is:
F'P
M=,;
aCos[t]+c
2
bSin[t]
2
The cartesian equation of the geometric place sought is:
In[]:=
Eliminatex==,y==,t
aCos[t]+c
2
bSin[t]
2
Out[]=
2
b
a
2
b
2
y
2
b
2
(-c+2x)
2
a
2
b
2
y
2
b
2
c
2
b
2
a
2
b
2
b
2
x
2
a
2
y
2
b
2
(-c+2x)
2
a
2
b
2
y
2
(2x-c)
2
a
4
2
y
2
b
2
x-
c
2
2
a
4
2
y
2
b
4
It’s an ellipse.
Let a = 4, b = 3, then c:
In[]:=
c=-
2
4
2
3
7
;2
x-
7
2
4
2
y
9
4
Method 2:
Let M(x, y) the midpoint, {,) a point on the ellipse and F’(c, 0) the focus, then:
P
1
x
1
y
1
x
1
2
x
1
y
1
2
y
1
Substituting in the ellipse’s equation :
2
(2x-c)
2
a
2
(2y)
2
b
Substituting a, b, c and simplifying:
2
x-
7
2
4
2
y
9
4
Note: For animation it is much easier to use parametric equations that involve trigonometric functions.
Animation:
In[]:=
Manipulatea=4;b=3;c=,;Show[{ParametricPlot[M[θ],{θ,0,θ1},ColorFunction->"BlueGreenYellow"],ParametricPlot[P[θ],{θ,0,2π},PlotStyle{Blue,Thick}],Graphics[{Black,PointSize[0.018],Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],Point[F'],Text[Style["F'",14,Bold],F',{0,-1.2}],Point[P[θ1]],Text[Style["P",14,Bold],P[θ1],{0,-1.2}],Point[M[θ1]],Text[Style["M",14,Bold],M[θ1],{0,-1.2}],Thick,Line[{P[θ1],F'}]}]},AspectRatioAutomatic,AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-4.5,4.5},{-3.5,3.5}}],Style["Geometric locus of M: Ellipse",Bold,Large],{{θ1,0.00002,"Value ()"},0.000001,2π,0.1},ControlPlacementTop
7
;O1={0,0};F'={c,0};P[θ_]:={aCos[θ],bSin[θ]};M[θ_]:=aCos[θ]+c
2
bSin[θ]
2
θ
1
Out[]=
