# How can I fill in a section of an ellipse converging on a focus rather than the center?

I am trying to plot a simple display of Kepler's second law - equal areas of an ellipse swept out in an equal amount of time. The problem is that I need my sections area sections to have a focus as their point of convergence, not the center. Is the problem caused by my use of parametric equations?

xe[t_] := 5*Cos[t];
ye[t_] := 3*Sin[t];
re[t_] := Sqrt[25*Cos[t]^2 + 9*Sin[t]^2]

P := {2.5, 0};
Q := {xe[Pi/6], ye[Pi/6]};
R := {xe[Pi/3], ye[Pi/3]};
S := {xe[7 Pi/6], ye[7 Pi/6]};
T := {xe[4 Pi/3], ye[4 Pi/3]}

p1 = ParametricPlot[{xe[t], ye[t]}, {t, 0, 2 Pi}, {r, 0, 1}];
l1 = ListLinePlot[{P, Q}];
l2 = ListLinePlot[{P, R}];
l3 = ListLinePlot[{P, S}];
l4 = ListLinePlot[{P, T}];
Show[p1, l1, l2, l3, l4]


Here is just a draft that needs to be updated later since I am not so familar with Physic. The code has some bugs :-(

Accoring to Kepler's Equation : $$M=E-\mathrm{e}\sin E$$

Clear["*"];
a = 5;
b = 3;
c = Sqrt[a^2 - b^2];
e = c/a;
p = b^2/c;
ρ[θ_] = (p*e)/(1 + e*Cos[θ]);
fig = PolarPlot[ρ[θ], {θ, 0, 2 π}];

angle[M_] :=
2 (π + ArcTan[Sqrt[(1 + e)/(1 - e)] Tan[eE/2]]) /.
NSolve[M == eE - e*Sin[eE] && 0 <= eE <= 2 π, eE] // First;
{θ1, θ2, θ3, θ4} =
angle /@ {.01*2 π, 0.11*2 π, 0.6*2 π, 0.7*2 π};

reg1 = ParametricRegion[ρ[θ]*
r*{Cos[θ],
Sin[θ]}, {{θ, θ1, θ2}, {r, 0, 1}}];
reg2 = ParametricRegion[ρ[θ]*
r*{Cos[θ],
Sin[θ]}, {{θ, θ3, θ4}, {r, 0, 1}}];
Show[fig, RegionPlot[reg1, PlotStyle -> Red],
RegionPlot[reg2, PlotStyle -> Yellow], PlotRange -> All]
`

• Fantastic! I can update the math to make if needed. – Earthling Ben Oct 3 at 2:11