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

*To*: mathgroup at smc.vnet.net*Subject*: [mg71761] Re: Area of ellipse between major axis and ray through focus, given angle*From*: "David W. Cantrell" <DWCantrell at sigmaxi.net>*Date*: Tue, 28 Nov 2006 06:04:15 -0500 (EST)*Organization*: NewsReader.Com Subscriber*References*: <ek96ij$fd5$1@smc.vnet.net>

"Kelly Jones" <kelly.terry.jones at gmail.com> wrote: > 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? Please see my article Area of a focal sector of an ellipse (sci.math, 2003 Apr. 2) <http://groups.google.com/group/sci.math/msg/911a2ab6c3e1450e>. > I'm guessing one of EllipticE/EllipticF/EllipticK gives the area as a > function of theta, but I can't figure out which one. As the above link shows, the area can be expressed in terms of elementary functions. > I also can't figure out what function gives theta as a function of the > area? There is no hope of expressing that in closed form using familiar functions, be they elementary or not. > Finally, for a fixed value of ec (eccentricity), what are the power > series expansions for the functions taking theta to area and vica > versa? For theta to area, use Series with the function given at the link; for "vice versa", use InverseSeries. David W. Cantrell