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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