Difference between revisions of "Ellipse"
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'''Ellipse''' | '''Ellipse''' | ||
| − | An ellipse is a [[conic section]] formed by cutting through a cone on an angle. More specifically, it is defined as the [[locus]], or set, of all points P such that the sum of the distances from P to the to foci ( | + | An ellipse is a [[conic section]] formed by cutting through a cone on an angle. More specifically, it is defined as the [[locus]], or set, of all points P such that the sum of the distances from P to the to foci (sing. [[focus]]) is a constant. Ellipses tend to resemble "flattened" circles. They occur in nature as well: 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 the two foci of the ellipse coincide. |
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| + | Assuming an ellipse is not a circle, there will wider from left to right or taller from the bottom to the top of the ellipse. 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'''. (a more precise definition needed) | ||
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To draw an ellipse with two pushpins, a rubber band, pencil and paper, stick the pushpins in the paper (these will be the "foci"), the rubber band around the pins, and trace out (please finish description, wording gets awkward after this for me). | To draw an ellipse with two pushpins, a rubber band, pencil and paper, stick the pushpins in the paper (these will be the "foci"), the rubber band around the pins, and trace out (please finish description, wording gets awkward after this for me). | ||
| − | 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>. |
| + | (definition of eccentricity and polar equation needed) | ||
| + | |||
| + | The three-dimensional counterpart of the ellipse is the [[ellipsoid]]. | ||
Revision as of 17:55, 19 June 2006
Ellipse
An ellipse is a conic section formed by cutting through a cone on an angle. More specifically, it is defined as the locus, or set, of all points P such that the sum of the distances from P to the to foci (sing. focus) is a constant. Ellipses tend to resemble "flattened" circles. They occur in nature as well: 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 the two foci of the ellipse coincide.
Assuming an ellipse is not a circle, there will wider from left to right or taller from the bottom to the top of the ellipse. 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. (a more precise definition needed)
To draw an ellipse with two pushpins, a rubber band, pencil and paper, stick the pushpins in the paper (these will be the "foci"), the rubber band around the pins, and trace out (please finish description, wording gets awkward after this for me).
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.
