ellipse (idea)

The ellipse is a slightly generalized form of the circle. A further generalization is an egg-shape. The generating equation for an ellipse with foci at (sqrt(a²-b²),0) and (-sqrt(a²-b²),0) having 'radius' 2*a is:

x²/a²+y²/b²=1

Below I have included a sketch of an ellipse.

                      ,....onOK@@@@@@@@@@@HQme....,
               ,..szSZSZF'`         |         `'TUXUXux..,
            ,zrP'`                                     `'Gcc,
         ,xw'` \'--,                |                      `'wx,
       .u'`     \  `--_                                      `'n.
     ,dy`        \     '--,         |                           `qb,
    /7`           \       `--_ r2   b                             `A\
   4y`           r1\          '--,  |                               \D
  ,I'               \            `--_                               `U,
  dp                 \              |'--,                            qb
 ,j'                  \                 `--_                         `t,
 AV                    \            |       '--,                      VA
 69- - - - - - - - - - -O- - - - - - - - - - - -O- -a- - - - - - - - -96
 VA                  (focus)        |        (focus)                  AV
 `t,                                                                 ,j'
  qb                                |                                dp
  `I,                                                               ,U'
   \D                               |                              4y`
    VA,                                                           /7`
     `qb,                           |                           ,dy`
       `'n.                                                   .u'`
         `'xw,                      |                      ,mx'`
            `'Gcc,                                     ,zzN'`
               ``'Tuxuxux.,         |         ,.szszszF'``
                      ````'TTOK@@@@@@@@@@@HQTT'````

The points on the edge of the ellipse can alse be described as those points p where r1+r2 = 2*a, with r1 measured as 'the distance from p to one focus' and r2 measured as 'the distance from p to the other focus'. (Unfortunately, the 'ellipse' above isn't quite right proportionally, but it should be enough to give a general idea.)

Some properties of the ellipse:

• Area: a*b*pi (note that when a=b, the ellipse becomes a circle and the area is a*a*pi)
• Circumference: unknown. Much numerical analysis must be done to approximate this value for any given a and b. See also elliptic integral.
• Letting x=a*cos(t) and y=b*sin(t) for real numbers t will also generate the (x,y) coordinates of the points on the edge of the ellipse.