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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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