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

*To*: mathgroup at smc.vnet.net*Subject*: [mg71939] Re: Area of ellipse between major axis and ray through focus, given angle*From*: rip pelletier <bitbucket at comcast.net>*Date*: Tue, 5 Dec 2006 06:04:37 -0500 (EST)*References*: <ek96ij$fd5$1@smc.vnet.net>

In article <ek96ij$fd5$1 at 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? > two books which i find very useful for doing orbital mechanics are: prussing & conway, "orbital mechanics", 1993, oxford. bate, mueller & white, "fundamentals of astrodynamics", 1971, dover. kepler's equation for position-on-orbit is pretty and simple; and a geometric proof of it leads to a a pretty and simple solution for the area. the key is to define an auxiliary angle called the eccentric anomaly, for which see mathworld. http://mathworld.wolfram.com/EccentricAnomaly.html especially, note the last sentence, which tells you how to get the area: "M can also be interpreted as the area of the shaded region in the above figure" vale, rip -- NB eddress is r i p 1 AT c o m c a s t DOT n e t