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]