Circle in the plane - Revision history

Vipul: /* General definition */

2011-07-29T23:24:59Z

General definition

← Older revision		Revision as of 23:24, 29 July 2011
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	* The group of all orthogonal motions fixing the origin, sends each circle centered at the origin, to itself		* The group of all orthogonal motions fixing the origin, sends each circle centered at the origin, to itself

	[[File:Circle.png\|~~800px~~]]		[[File:Circle.png\|400px]]

	===Variant definitions===		===Variant definitions===

Vipul: /* Definition */

2011-07-29T23:24:40Z

Definition

← Older revision		Revision as of 23:24, 29 July 2011
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	* Given two circles of the same radius but with different centers, there is a translation of <math>\R^2</math> that sends one circle to the other. Namely, choose the translation that sends the center of the first circle, to the center of the second circle.		* Given two circles of the same radius but with different centers, there is a translation of <math>\R^2</math> that sends one circle to the other. Namely, choose the translation that sends the center of the first circle, to the center of the second circle.
	* The group of all orthogonal motions fixing the origin, sends each circle centered at the origin, to itself		* The group of all orthogonal motions fixing the origin, sends each circle centered at the origin, to itself

			[[File:Circle.png\|800px]]

	===Variant definitions===		===Variant definitions===

Vipul: /* Families of circles */

2011-07-24T22:13:05Z

Families of circles

← Older revision		Revision as of 22:13, 24 July 2011
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	* [[Family of concentric circles]]		* [[Family of concentric circles]]
	* [[Family of coaxial circles with intersection points]]		* [[Family of coaxial circles with intersection points]]
	* [{Family of coaxial circles with limit points]]		* [[Family of coaxial circles with limit points]]

	==Related notions==		==Related notions==

Vipul: 3 revisions

2008-05-18T19:34:42Z

3 revisions

← Older revision	Revision as of 19:34, 18 May 2008
(No difference)

Vipul at 16:18, 4 April 2008

2008-04-04T16:18:39Z

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 {{fillin}}
 ==Related notions==
 * [[Circle on a surface]]
 * [[Circle on the 2-sphere]]

Vipul at 16:01, 4 April 2008

2008-04-04T16:01:13Z

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 In complex analysis, it is sometimes convenient to view a line as a circle. We think of the center of the line as being a point at infinity, and the radius as infinity. This makes the theory of inversion, the geometric intuition behind complex analysis, as well as coaxial systems of circles, easier to comprehend.
 ==Related notions==
 * [[Circle on a surface]]
 * [[Circle on the 2-sphere]]

Vipul: New page: {{quotation|''A generalization to higher dimensions is sphere in Euclidean space''}} ==Definition== ===General definition=== Consider $\R^2$ , the Euclidean plane. Let...

2008-04-03T23:10:01Z

New page: {{quotation|''A generalization to higher dimensions is sphere in Euclidean space''}} ==Definition== ===General definition=== Consider <math>\R^2</math>, the Euclidean plane. Let...

New page

{{quotation|''A generalization to higher dimensions is [[sphere in Euclidean space]]''}}

==Definition==

===General definition===

Consider <math>\R^2</math>, the [[Euclidean plane]]. Let <math>p \in \R^2</math> be a point and <math>r > 0</math> be a positive real number. The circle with center <math>p</math> and radius <math>r</math> is the set of all points in <math>\R^2</math> that have distance exactly <math>r</math> from <math>p</math>.

Some easy facts:

* Two circles centered at the same point are termed concentric circles. Given two concentric circles, there is a dilation, or scaling, about the common center that takes one circle to the other
* Given two circles of the same radius but with different centers, there is a translation of <math>\R^2</math> that sends one circle to the other. Namely, choose the translation that sends the center of the first circle, to the center of the second circle.
* The group of all orthogonal motions fixing the origin, sends each circle centered at the origin, to itself

===Variant definitions===

In complex analysis, it is sometimes convenient to view a line as a circle. We think of the center of the line as being a point at infinity, and the radius as infinity. This makes the theory of inversion, the geometric intuition behind complex analysis, as well as coaxial systems of circles, easier to comprehend.

==Related notions==

* [[Circle on a surface]]
* [[Circle on the 2-sphere]]