Area of ellipse between major axis and ray through focus, given angle

• To: mathgroup at smc.vnet.net
• Subject: [mg71601] Area of ellipse between major axis and ray through focus, given angle
• From: "Kelly Jones" <kelly.terry.jones at gmail.com>
• Date: Sat, 25 Nov 2006 05:36:36 -0500 (EST)

```Given:

1) an ellipse with eccentricity "ec", one focus on the origin, and
the major axis along the x-axis

2) a ray through the origin at angle theta to the x-axis

Question:

What Mathematica function gives the relation/inverse relation between
the angle theta and the area of the ellipse between the x-axis and the ray?

I'm guessing one of EllipticE/EllipticF/EllipticK gives the area as a
function of theta, but I can't figure out which one.

I also can't figure out what function gives theta as a function of the area?

Finally, for a fixed value of ec (eccentricity), what are the power
series expansions for the functions taking theta to area and vica
versa?

Ugly drawing:

(* numbers chosen "randomly" for drawing purposes only *)
lower = ParametricPlot[{x,-(Sqrt[3]*Sqrt[3 - 8*x - 16*x^2])/8}, {x,-3/4,1/4}]
upper = ParametricPlot[{x, (Sqrt[3]*Sqrt[3 - 8*x - 16*x^2])/8}, {x,-3/4,1/4}]
line = Line[{{0,0},{.14,.27097}}]
arc = Circle[{0,0},.05,{0,1.09391}]

Show[lower,upper,Graphics[line],Graphics[arc],
AspectRatio->Automatic,Ticks->None]

```

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