Difference between revisions of "Ellipse"
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Ellipses tend to resemble [[circle]]s which have been "flattened" or "stretched." They occur in nature as well as in mathematics: as was proven in [[Kepler's Laws]], the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci. Note that the circle is just a special case of the ellipse, just as a square is to a rectangle, and occurs when (in the first definition) the cut is [[perpendicular]] to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide. | Ellipses tend to resemble [[circle]]s which have been "flattened" or "stretched." They occur in nature as well as in mathematics: as was proven in [[Kepler's Laws]], the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci. Note that the circle is just a special case of the ellipse, just as a square is to a rectangle, and occurs when (in the first definition) the cut is [[perpendicular]] to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide. | ||
| − | For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat." The segment connecting the center of the ellipse to one of the "farther away ends" is called the ''semimajor axis'' and the segment connecting the center to a closer end is called the ''semiminor axis''. These two segments | + | For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat." The segment connecting the center of the ellipse to one of the "farther away ends" is called the ''[[semimajor axis]]'' and the segment connecting the center to a closer end is called the ''[[semiminor axis]]''. These two segments are perpendicular. Drawing all four semi-axes divides the ellipse into 4 [[congruent (geometry)|congruent]] quarters. |
| − | To draw an ellipse with two pushpins, a rubber band, pencil, and paper, stick the pushpins in the paper (these will be the "foci") | + | An ellipse may easily be constructed with common household materials, using the second definition of an ellipse. To draw an ellipse with two pushpins, a rubber band, pencil, and paper, stick the pushpins in the paper (these will be the "foci") and wrap the rubber band around the pins. Hold the pencil on the paper such that the rubber band is taut around the pins and pencil tip. Then move the pencil tip while keeping the rubber band taut. This traces out an ellipse, since the distance between the pencil tip and the two foci remains constant (because of the rubber band). |
The general equation of an ellipse with semi-minor and -major axes a and b and center C(h,k) is <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>, or, when centered at the origin, <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>. | The general equation of an ellipse with semi-minor and -major axes a and b and center C(h,k) is <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>, or, when centered at the origin, <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>. | ||
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The three-dimensional counterpart of the ellipse is the [[ellipsoid]]. | The three-dimensional counterpart of the ellipse is the [[ellipsoid]]. | ||
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| + | ==See also== | ||
| + | * [[Parabola]] | ||
| + | * [[Conic section]]s | ||
| + | * [[Geometry]] | ||
| + | * [[Polynomial]]s | ||
Revision as of 21:09, 7 January 2007
An ellipse is a conic section formed by cutting through a cone at an angle. Equivalently, it is defined as the locus, or set, of all points 
such that the sum of the distances from to two fixed foci (singular focus) is a constant. (The equivalence of these two definitions is a non-trivial fact.)
Ellipses tend to resemble circles which have been "flattened" or "stretched." They occur in nature as well as in mathematics: as was proven in Kepler's Laws, the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci. Note that the circle is just a special case of the ellipse, just as a square is to a rectangle, and occurs when (in the first definition) the cut is perpendicular to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.
For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat." The segment connecting the center of the ellipse to one of the "farther away ends" is called the semimajor axis and the segment connecting the center to a closer end is called the semiminor axis. These two segments are perpendicular. Drawing all four semi-axes divides the ellipse into 4 congruent quarters.
An ellipse may easily be constructed with common household materials, using the second definition of an ellipse. To draw an ellipse with two pushpins, a rubber band, pencil, and paper, stick the pushpins in the paper (these will be the "foci") and wrap the rubber band around the pins. Hold the pencil on the paper such that the rubber band is taut around the pins and pencil tip. Then move the pencil tip while keeping the rubber band taut. This traces out an ellipse, since the distance between the pencil tip and the two foci remains constant (because of the rubber band).
The general equation of an ellipse with semi-minor and -major axes a and b and center C(h,k) is 
, or, when centered at the origin, 
.
(definition of eccentricity and polar equation needed)
The three-dimensional counterpart of the ellipse is the ellipsoid.
