# Difference between revisions of "Ellipse"

m (subcat) |
(username removed) (Tidy up, added a bit about areas and circumferences) |
||

(2 intermediate revisions by one other user not shown) | |||

Line 1: | Line 1: | ||

[[Image:658px-Elipse svg.png|right|thumb|400px|In the figure, ''a'' is the semi-major axis, ''b'' is the semi-minor axis, F1 and F2 are the two focal points. The distance F1-X-F2 is constant.]] | [[Image:658px-Elipse svg.png|right|thumb|400px|In the figure, ''a'' is the semi-major axis, ''b'' is the semi-minor axis, F1 and F2 are the two focal points. The distance F1-X-F2 is constant.]] | ||

− | An '''ellipse''' is a geometric figure that looks like a squashed [[circle]]. The more squashed it is, the greater its | + | An '''ellipse''' is a geometric figure that looks like a squashed [[circle]]. The more squashed it is, the greater its eccentricity. |

− | It is defined as the set of all points in a plane where the sum of the distance from two points, the ''foci'', is the same. A special case of the ellipse is the circle, where the two points are coincident. | + | It is defined as the set of all points in a plane where the sum of the distance from two points, the ''foci'', is the same. A special case of the ellipse is the circle, where the two points are coincident. A circle has an eccentricity of 0. |

To construct an ellipse, place tacks at two points on a piece of paper with a string between them. Trace out the curve of the ellipse with a pencil which is always pushing against the string. The longer the string, the less eccentric the ellipse will be (and more circular). | To construct an ellipse, place tacks at two points on a piece of paper with a string between them. Trace out the curve of the ellipse with a pencil which is always pushing against the string. The longer the string, the less eccentric the ellipse will be (and more circular). | ||

Line 15: | Line 15: | ||

The terms ''eccentric'' and ''circular'' are antonyms. | The terms ''eccentric'' and ''circular'' are antonyms. | ||

− | A pair of line segments drawn from one | + | A pair of line segments drawn from one focal point of an ellipse to the curve and from there to its other focal point form an angle whose bifurcating line is always perpendicular to the tangent of the curve. This feature of an ellipse has been exploited in rooms having elliptical walls or (more dramatically) elliptical ceilings. |

− | A whisper at one focal point is easily heard at | + | A whisper at one focal point is easily heard at the other focal point, because the sound waves bounce off the walls and combine again at the other focal point. |

− | Ellipses became important in [[astronomy]] in the early 1600s, when [[Kepler]] proved that planets revolving the sun always follow elliptical [[orbit]]s, with the sun at one of the foci. This helped overturn the [[Ptolemaic theory]], and led to [[Issac Newton]]'s [[law of gravitation]]. | + | Ellipses became important in [[astronomy]] in the early 1600s, when [[Kepler]] proved that planets revolving the sun always follow elliptical [[orbit]]s, with the sun at one of the foci. This helped overturn the [[Ptolemaic theory]], and led to [[Newton|Issac Newton]]'s [[Gravitation|law of gravitation]]. |

− | [[ | + | ==Area and Circumference== |

+ | |||

+ | Unlike a [[circle]], there is no closed form for the [[circumference]] of an ellipse. The [[area]] of an ellipse is given by: | ||

+ | |||

+ | <math>A = \pi ab</math> | ||

+ | |||

+ | [[Category:Plane Geometry]] |

## Latest revision as of 11:11, 24 September 2016

An **ellipse** is a geometric figure that looks like a squashed circle. The more squashed it is, the greater its eccentricity.

It is defined as the set of all points in a plane where the sum of the distance from two points, the *foci*, is the same. A special case of the ellipse is the circle, where the two points are coincident. A circle has an eccentricity of 0.

To construct an ellipse, place tacks at two points on a piece of paper with a string between them. Trace out the curve of the ellipse with a pencil which is always pushing against the string. The longer the string, the less eccentric the ellipse will be (and more circular).

The general algebraic formula for an ellipse is given by:

An ellipse is a conic section, the intersection of a plane and a cone, where the angle of the plane to the cone's axis is greater than the angle of the cone with its axis.

The terms *eccentric* and *circular* are antonyms.

A pair of line segments drawn from one focal point of an ellipse to the curve and from there to its other focal point form an angle whose bifurcating line is always perpendicular to the tangent of the curve. This feature of an ellipse has been exploited in rooms having elliptical walls or (more dramatically) elliptical ceilings. A whisper at one focal point is easily heard at the other focal point, because the sound waves bounce off the walls and combine again at the other focal point.

Ellipses became important in astronomy in the early 1600s, when Kepler proved that planets revolving the sun always follow elliptical orbits, with the sun at one of the foci. This helped overturn the Ptolemaic theory, and led to Issac Newton's law of gravitation.

## Area and Circumference

Unlike a circle, there is no closed form for the circumference of an ellipse. The area of an ellipse is given by: