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en>Msrasnw: /* References */ lining to L. Bostock, and S. Chandler

2014-12-06T15:19:55Z

References: lining to L. Bostock, and S. Chandler

← Older revision		Revision as of 17:19, 6 December 2014
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	~~{{no footnotes\|date=October 2013}}~~		I am Shantae from Bottrop Altstadt. I am learning to play the Pedal Steel Guitar. Other hobbies are Homebrewing.<br><br>Feel free to surf to my web page :: [http://tempstreets.com/wiki/index.php?title=User:BookerT92qxmixl Basement Windows]
	In [[Geometry]], a '''locus''' (plural: ''loci'') is a set of points whose location satisfies or is determined by one or more specified conditions i.e., 1)every point satisfies a given condition and 2)every point satisfiying it is in that particular locus
	[[File:Locus Curve.svg\|thumb\|right\|400px\|Each curve in this example is the ''locus'' of a set of points that lie on any line defined as the [[Conchoid (mathematics)\|conchoid]] of a circle centered at point ''P'' and the line ''l''. In this example, ''P'' is 7cm from ''l''.]]

	~~==Commonly studied loci==~~
	~~Examples from plane geometry:~~
	* The set of points equidistant from two points is a [[perpendicular bisector]] to the [[line segment]] connecting the two points.

	* The set of points equidistant from two lines which cross is the [[angle bisector]].

	* All [[conic section]]s are loci:
	** [[Parabola]]: the set of points equidistant from a single point (the [[focus (geometry)\|focus]]) and a line (the [[directrix (conic section)\|directrix]]).
	** [[Circle]]: the set of points for which the distance from a single point is constant (the [[radius]]). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is not 1) is referred to as a [[Circles of Apollonius\|Circle of Apollonius]].
	**[[Hyperbola]]: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
	**[[Ellipse]]: the set of points for each of which the sum of the distances to ~~two given foci is a constant. In particular,~~ the ~~[[circle]] is a locus~~.

	~~==Proof of a Locus==~~
	~~In order to prove the correctness of a locus, one generally divides the proof into two stages:~~
	* Proof that all the points that satisfy the conditions are ~~on the locus.~~
	* Proof that all the points on the locus satisfy the conditions.

	~~== Examples ==~~
	~~[[File:Locus apollonius.svg\|thumb\|(distance PA) = 3.(distance PB)]]~~
	~~The locus of the points P that have a given ratio of distances k = d1/d2 to two given points~~.<br>
	~~In this example we choose k = 3 , A(-1,0) and B(0,2) as the fixed points.~~

	~~:: P(x,y) is a point of the locus~~
	: <~~math~~>~~\Leftrightarrow \|PA\| = 3 \|PB\| </math>~~
	: ~~<math> \Leftrightarrow \|PA\|^2 = 9 \|PB\|^2 </math>~~
	~~:<math>\Leftrightarrow (x+1)^2+(y-0)^2=9(x-0)^2+9(y-2)^2 </math>~~
	: ~~<math> \Leftrightarrow 8(x^2+y^2)-2x-36y+35 =0 </math>~~
	~~:<math>\Leftrightarrow \left(x-\frac18\right)^2+\left(y-\frac94\right)^2=\frac{45}{64}</math>~~

	~~This equation represents a [[circle]] with center (1/8,9/4) and radius <math> \frac{3}{8}\sqrt{5}</math>.~~

	[~~[File~~:~~Locus3a.svg\|thumb\| Locus of point C?]]~~
	~~A triangle ABC has a fixed side [AB] with length c.~~
	~~We determine the locus of the third [[Vertex (geometry)\|vertex]] C such that~~
	~~the [[Median (geometry)\|medians]] from A en C are [[orthogonal]].~~

	~~We choose an [[orthonormal]] [[coordinate system]] such that A(-c~~/~~2,0), B(c~~/~~2,0)~~.
	~~C(x,y) is the variable third vertex. The center of [BC] is M( (2x+c)~~/~~4, y~~/~~2 )~~. ~~The median from C has a slope y/x. The median AM has a [[slope]] 2y/(2x+3c).~~

	~~[[File:Locus3.svg\|thumb\|The locus is a circle]]~~
	~~::C(x,y) is a point of the locus~~
	~~:<math>\Leftrightarrow</math> The medians from A and C are orthogonal~~
	~~:<math>\Leftrightarrow \frac{y}{x} \cdot \frac{2y}{2x+3c} = -1 </math>~~
	~~:<math>\Leftrightarrow 2 y^2 + 2x^2 + 3c x = 0 </math>~~
	~~:<math>\Leftrightarrow x^2 + y^2 + (3c/2) x = 0 </math>~~
	~~:<math>\Leftrightarrow (x + 3c/4)^2 + y^2 = 9c^2/16 </math>~~

	~~The locus of the vertex C is a circle with center (-3c/4,0) and radius 3c/4.~~

	~~[[File:Geassocieerde rechten.svg\|thumb\|The intersection point of the associated lines k and l describes the circle]]~~
	~~A locus can also be defined by two associated curves depending on one common [[parameter]].~~
	~~If the parameter varies, the intersection points of the associated curves describe the locus.~~

	On the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is [[perpendicular]] to k. The angle between k and m is the parameter.
	k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.

	~~==See also==~~

	~~{{Empty section\|date=January 2014}}~~

	==~~References==~~
	~~{{Reflist}}~~
	*Robert Clarke James, Glenn James: ''Mathematics Dictionary''. Springer 1992, ISBN 978-0-412-99041-0, p. 255 ({{Google books\|UyIfgBIwLMQC\|restricted online copy\|page=255}})
	*[[Alfred North Whitehead]]: ''An Introduction to Mathematics''. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-19784-2, pp. 121 ({{Google books\|UyIfgBIwLMQC\|restricted online copy\|page=121}})
	*George Wentworth: ''Junior High School Mathematics: Book III''. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-15236-0, pp. 265 ({{Google books\|cPlTB4qe40MC\|restricted online copy\|page=265}})

	~~[[Category~~:~~Elementary geometry]~~]

en>Clarkcj12: Reverted 1 edit by 108.202.108.134 identified as test/vandalism using STiki

2013-12-03T01:29:09Z

Reverted 1 edit by 108.202.108.134 identified as test/vandalism using STiki

← Older revision		Revision as of 03:29, 3 December 2013
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	~~It begins~~ by ~~consuming~~ in a ~~pair~~ of ~~one~~'~~s lovely island where your peaceful village is in beaches~~ and ~~woods up until~~ the ~~enemies known since the BlackGuard led by Lieutenant Hammerman invades your snowdonia~~. ~~After managing to guard against a small bit of invasion force~~, ~~he proposes to avenge his loss appearing in battle~~.~~<br><br>To comprehend coins and gems, will need~~ to ~~obtain~~ the ~~Clash towards Clans hack equipment by clicking on~~ the ~~earn button~~. ~~Contingent by~~ the ~~operating framework that happen to be utilizing, you will market~~ the ~~downloaded document as~~ the ~~admin. Furnish~~ a [~~http://www~~.~~Twitpic.com/tag/person%27s+log person's log~~] ~~in Id and select~~ the ~~gadget~~. ~~Immediately after this, you are get into~~ the ~~quantity~~ of ~~gems or coins~~ that ~~if you want~~ to ~~and start off unquestionably~~ the ~~Clash~~ of ~~Clans identify instrument.<br><br>Interweaving social trends form a formidable net in~~ which ~~we're all trapped. When Each Tygers~~ of ~~Pan Tang sang 'It's lonely towards~~ the ~~top~~. ~~Everyones trying to do families in', these people acquired much from clash~~ of ~~clans chop tool no survey. A society without conflict~~ of ~~clans hack tool no survey~~ is ~~like~~ a ~~society with no knowledge~~, ~~in that it~~ is ~~quite good~~.~~<br><br>Battle~~ of ~~Clans is without any doubt~~ a ~~popular sport designed~~ to ~~end up being experienced~~ on ~~multiple systems, a reputable iOS and also google android~~. The ~~overall game is terribly intriguing but presently presently comes~~ a ~~spot on legend~~, ~~where~~ the ~~device gets trapped because because of not enough gems~~. ~~However~~, ~~this problem~~ is ~~now able to easily end up resolved.~~<br><br>~~If you are looking Conflict of The entire family Jewels Free, or may well just buying~~ a ~~Accommodement Conflict of Tribes~~, ~~right now the smartest choice on your internet, absolutely free as well as only takes a short while to get all many.~~<br><br>~~Via borer on~~ a ~~boondocks anteroom you possibly is able to appearance added advice~~ with ~~regard to~~ that ~~play~~, ~~scout~~, ~~accord troops~~, ~~or attack~~. ~~Of course~~, ~~these positive results will rely on all that appearance~~ of the ~~war you might be found in.~~<br><br>A ~~leap into~~ the ~~pre~~-~~owned or operated xbox sports marketplace~~. ~~Several players will get a Battle~~ of ~~Clans Hack~~ and ~~complete this game really really fast. Several shops let these gaming titles being dealt in following which promote them at your lessened cost. Might~~ be by ~~far~~ one of the ~~most cost-effective technique~~ to ~~be newer video games~~ the ~~particular higher cost Should you loved this article~~ and ~~you would want to receive more information regarding how to hack clash~~ of ~~clans~~ - [~~http~~:~~//circuspartypanama~~.~~com circuspartypanama~~.~~com]~~, ~~i implore you to visit our own web site.~~ .		{{no footnotes\|date=October 2013}}
			In [[Geometry]], a '''locus''' (plural: ''loci'') is a set of points whose location satisfies or is determined by one or more specified conditions i.e., 1)every point satisfies a given condition and 2)every point satisfiying it is in that particular locus
			[[File:Locus Curve.svg\|thumb\|right\|400px\|Each curve in this example is the ''locus'' of a set of points that lie on any line defined as the [[Conchoid (mathematics)\|conchoid]] of a circle centered at point ''P'' and the line ''l''. In this example, ''P'' is 7cm from ''l''.]]

			==Commonly studied loci==
			Examples from plane geometry:
			* The set of points equidistant from two points is a [[perpendicular bisector]] to the [[line segment]] connecting the two points.

			* The set of points equidistant from two lines which cross is the [[angle bisector]].

			* All [[conic section]]s are loci:
			** [[Parabola]]: the set of points equidistant from a single point (the [[focus (geometry)\|focus]]) and a line (the [[directrix (conic section)\|directrix]]).
			** [[Circle]]: the set of points for which the distance from a single point is constant (the [[radius]]). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is not 1) is referred to as a [[Circles of Apollonius\|Circle of Apollonius]].
			**[[Hyperbola]]: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
			**[[Ellipse]]: the set of points for each of which the sum of the distances to two given foci is a constant. In particular, the [[circle]] is a locus.

			==Proof of a Locus==
			In order to prove the correctness of a locus, one generally divides the proof into two stages:
			* Proof that all the points that satisfy the conditions are on the locus.
			* Proof that all the points on the locus satisfy the conditions.

			== Examples ==
			[[File:Locus apollonius.svg\|thumb\|(distance PA) = 3.(distance PB)]]
			The locus of the points P that have a given ratio of distances k = d1/d2 to two given points.<br>
			In this example we choose k = 3 , A(-1,0) and B(0,2) as the fixed points.

			:: P(x,y) is a point of the locus
			: <math>\Leftrightarrow \|PA\| = 3 \|PB\| </math>
			: <math> \Leftrightarrow \|PA\|^2 = 9 \|PB\|^2 </math>
			:<math>\Leftrightarrow (x+1)^2+(y-0)^2=9(x-0)^2+9(y-2)^2 </math>
			: <math> \Leftrightarrow 8(x^2+y^2)-2x-36y+35 =0 </math>
			:<math>\Leftrightarrow \left(x-\frac18\right)^2+\left(y-\frac94\right)^2=\frac{45}{64}</math>

			This equation represents a [[circle]] with center (1/8,9/4) and radius <math> \frac{3}{8}\sqrt{5}</math>.

			[[File:Locus3a.svg\|thumb\| Locus of point C?]]
			A triangle ABC has a fixed side [AB] with length c.
			We determine the locus of the third [[Vertex (geometry)\|vertex]] C such that
			the [[Median (geometry)\|medians]] from A en C are [[orthogonal]].

			We choose an [[orthonormal]] [[coordinate system]] such that A(-c/2,0), B(c/2,0).
			C(x,y) is the variable third vertex. The center of [BC] is M( (2x+c)/4, y/2 ). The median from C has a slope y/x. The median AM has a [[slope]] 2y/(2x+3c).

			[[File:Locus3.svg\|thumb\|The locus is a circle]]
			::C(x,y) is a point of the locus
			:<math>\Leftrightarrow</math> The medians from A and C are orthogonal
			:<math>\Leftrightarrow \frac{y}{x} \cdot \frac{2y}{2x+3c} = -1 </math>
			:<math>\Leftrightarrow 2 y^2 + 2x^2 + 3c x = 0 </math>
			:<math>\Leftrightarrow x^2 + y^2 + (3c/2) x = 0 </math>
			:<math>\Leftrightarrow (x + 3c/4)^2 + y^2 = 9c^2/16 </math>

			The locus of the vertex C is a circle with center (-3c/4,0) and radius 3c/4.

			[[File:Geassocieerde rechten.svg\|thumb\|The intersection point of the associated lines k and l describes the circle]]
			A locus can also be defined by two associated curves depending on one common [[parameter]].
			If the parameter varies, the intersection points of the associated curves describe the locus.

			On the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is [[perpendicular]] to k. The angle between k and m is the parameter.
			k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.

			==See also==

			{{Empty section\|date=January 2014}}

			==References==
			{{Reflist}}
			*Robert Clarke James, Glenn James: ''Mathematics Dictionary''. Springer 1992, ISBN 978-0-412-99041-0, p. 255 ({{Google books\|UyIfgBIwLMQC\|restricted online copy\|page=255}})
			*[[Alfred North Whitehead]]: ''An Introduction to Mathematics''. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-19784-2, pp. 121 ({{Google books\|UyIfgBIwLMQC\|restricted online copy\|page=121}})
			*George Wentworth: ''Junior High School Mathematics: Book III''. BiblioBazaar LLC 2009 (reprint), ISBN 978-1-103-15236-0, pp. 265 ({{Google books\|cPlTB4qe40MC\|restricted online copy\|page=265}})

			[[Category:Elementary geometry]]

71.199.125.210: Revert to older version

2012-08-02T02:58:57Z

Revert to older version

New page

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