ellipse

The locus of points in the plane with some common sum-of-distances to 2 points:

{ P : d(A,P) + d(B,P) = 2R }

for points A,B in the plane. A and B are known as the foci (sing. focus) of the ellipse.

To draw an ellipse, take a sheet of paper and stick 2 thumbtacks through A and B. Now tie a piece of string into a loop. Slip the loop around the 2 thumbtacks, and put a pen in it. Trace that pen around the tacks while holding the string taut. It's obvious the result has to be an ellipse.

When defining a geometric object as a "locus" of points based based upon a metric, it is always interesting to consider what sorts of figures appear when applying other metrics than the well-known Pythagorean metric.

For example, when applying the "Manhattan" or taxicab metric to R², the locus definition of an "ellipse" always gives us an octagon.  For example, in the following diagram,

. . . . . . . . . . .
. . . . ._._. . . . .
. . . ./. . .\. . . .
. . ./. . . . .\. . .
. ./. . . . o . .\. .
. | . . . . . . . | .
. | . . . . . . . | .
. | . . o . . . . | .
. .\. . . . . . ./. .
. . .\. . . . ./. . .
. . . .\._._./. . . .
. . . . . . . . . . .

The foci are the two o characters.  The distance between the two foci (5) and the two distances from both foci to any point on the octagonal figure always add up to 16.

These octagonal ellipses have some very odd traits:

• All four diagonal sides are always the same length.
• The two vertical sides are always the same length as the "y" part of the distance between the two foci.
• The two horizontal sides are always the same length as the "x" part of the distance between the two foci.
• Since a circle is an ellipse whose foci are coincident, a "circle" using our metric is a square turned 45 degrees.  Talk about squaring the circle!

An ellipse defined using the box metric is still an octagon, but this time, the horizontal and vertical sides are the same length, and the diagonal sides grow as the foci move apart from each other.  A "circle" using the box metric looks like a rectangle.   See if you can draw the ellipse using the same two foci and total distance as above, but using the box metric instead.  Hint: the foci are 3 units apart this time. Answer

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.

Useful information about ellipses:
(note: a is the length of the major semiaxis, b is the length of the minor semiaxis, c is the distance from the focus to the center, and a²-b²=c²)

			Horizontal major axis		Horizontal minor axis
Standard Form:		(x-h)²/a²+(y-k)²/b²=1		(x-h)²/b²+(y-k)²/a²=1
Eccentricity:		e=c/a (e<1, e=0 ⇒ circle)	e=c/a (e<1, e=0 ⇒ circle)
Length of latera recta:	l=2b²/a				l=2b²/a 
Directrices:		x=±a²/c+h			y=±a²/c+k
Vertices:		(h±a,k±b)			(h±b,k±a)
Foci:			(h±c,k)				(h,k±c)
Parametric Form:	x′=asin(t) or	x′=acos(t)	x′=bsin(t) or	x′=bcos(t)
			y′=bcos(t) 	y′=bsin(t)	y′=acos(t)	y′=asin(t)

See also:

• Conic Section
• Parabola
• Hyperbola
• Circle

El*lipse" (?), n. [Gr. , prop., a defect, the inclination of the ellipse to the base of the cone being in defect when compared with that of the side to the base: cf. F. ellipse. See Ellipsis.]

1. Geom.

An oval or oblong figure, bounded by a regular curve, which corresponds to an oblique projection of a circle, or an oblique section of a cone through its opposite sides. The greatest diameter of the ellipse is the major axis, and the least diameter is the minor axis. See Conic section, under Conic, and cf. Focus.

2. Gram.

Omission. See Ellipsis.

3.

The elliptical orbit of a planet.

The Sun flies forward to his brother Sun; The dark Earth follows wheeled in her ellipse. Tennyson.

 

© Webster 1913.