en>IronGargoyle: Reverted edits by 121.54.58.145 (talk) to last revision by 94.79.52.163 (HG)

2014-04-25T11:54:16Z

Reverted edits by 121.54.58.145 (talk) to last revision by 94.79.52.163 (HG)

94.79.52.163: /* Problems with massive sample sizes */

2013-12-23T10:55:39Z

Problems with massive sample sizes

← Older revision		Revision as of 12:55, 23 December 2013
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	~~Games can give you~~ a ~~major universe of experience~~, ~~relaxation~~ and ~~exhilaration~~. ~~Someone can learn~~, ~~get~~ a ~~comman sense~~ of ~~success or merely enjoy beating down specific bad-guy~~. ~~No appear form~~ of ~~video video game players you~~ are ~~into~~, ~~add~~ the ~~helpful tips in~~ the ~~foregoing post to give your~~ [~~http://data.gov.uk/data/search?q=mid-section mid-section~~] ~~more fun whenever a play your next video game title Website~~.~~<br><br>~~		[[Image:Excentricidad.svg\|thumb\|Point F is a focus point for the red ellipse, green parabola and blue hyperbola.]]
			In [[geometry]], the '''foci''' ({{IPAc-en\|ˈ\|f\|oʊ\|s\|aɪ}}; singular '''focus''') are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining [[conic section]]s, the four types of which are the [[circle]], [[ellipse]], [[parabola]], and [[hyperbola]]. In addition, foci are used to define the [[Cassini oval]] and the [[Cartesian oval]].

	~~Cultivate~~ a ~~gaming program to deal~~ with. ~~Similar to required assignments time~~, ~~this video game program will improve manage~~ a ~~child's procedures~~. ~~When~~ the ~~times have always been~~ set~~, stick~~ to ~~ones schedule~~. ~~Do Not back~~ as a ~~product~~ of ~~whining or selling~~. ~~The schedule~~ is ~~only efficient~~ if ~~you just follow-through~~.~~<br><br>Temperance~~ is ~~essential~~ in ~~nearly things~~, and ~~enjoying clash~~ of ~~clans cheats~~ is ~~no different~~. ~~If you have any inquiries concerning in which~~ and ~~how~~ to ~~use~~ [~~http://circuspartypanama~~.~~com clash~~ of ~~clans hack free download]~~, ~~you can call us~~ at ~~our own web site~~. ~~Playing for hours on finish~~ is ~~not really good~~-~~for~~-~~you~~, ~~bodily or~~ in ~~your head~~. ~~There~~ are ~~some games out typically there~~ which ~~know this and will include measures~~ to ~~remind one to take rests~~. ~~Consider~~ the ~~initiative yourself~~, ~~although! Place~~ an ~~alarm so you don't play~~ for ~~over an hour right~~.~~<br><br>Consumers are~~ a ~~group of coders that loves with regard~~ to ~~play Cof~~. ~~We are continuously developing Cheats to speed up Levelling easily~~ and ~~to produce more gems for price~~. ~~Without our hacks it may well take you ages~~ to ~~reach your level~~.<br><br>~~Exclusive~~ some ~~online games provde~~ the ~~comfort~~ of ~~putting together~~ a ~~true-entire world timepiece accessible in the movie game itself. This particular really~~ is ~~usually~~ a ~~downside~~ to ~~full-monitor game titles~~. ~~You don~~'~~t want one using up even other~~ of ~~your time along with energy than within your ultimate budget place~~ a ~~moment~~ in ~~time clock~~ of ~~your similar to your display movie screen to be able if you want to monitor just how you~~'~~ve been enjoying~~.<br><br>~~When you need~~ to ~~defeat higher-level villages~~, ~~job aids you to make use~~ of a ~~mixture of troops not unlike Barbarians plus Archers plus those suicide wall bombers~~ to ~~bust down filters~~. ~~Goblins~~ can ~~also~~ be ~~a useful feature~~ the ~~combo simply considering that they attack different buildings~~. ~~You should understand if you would like~~ to ~~begin worrying about higher troops when your family can~~''~~t win battles offering Barbarians~~.<br><br>~~Disclaimer: I aggregate~~ the ~~useful information on this commodity by business a lot of CoC~~ and ~~accomplishing some seek out~~. To the ~~best involving my knowledge~~, is ~~it authentic along with I accept amateur costed all abstracts and computations~~. ~~Nevertheless, it is consistently accessible that i accept fabricated a aberration about or which a bold has afflicted rear publication~~. ~~Use as part of your very own risk, I don't accommodate virtually any warranty specifics~~. ~~Please get in blow if you'll acquisition annihilation amiss~~.		==Conics in geometry==

			===Defining conics in terms of two foci===

			An ellipse can be defined as the [[Locus (mathematics)\|locus]] of points for each of which the sum of the distances to two given foci is a constant.

			A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the [[Circles of Apollonius\|circle of Apollonius]], in terms of two different foci, as the set of points having a fixed ratio of distances to the two foci.

			A parabola is a limiting case of an ellipse in which one of the foci is a [[point at infinity]].

			A hyperbola can be defined as the locus of points for each of which the absolute value of the difference between the distances to two given foci is a constant.

			===Defining conics in terms of a focus and a directrix===

			It is also possible to describe all the conic sections in terms of a single focus and a single [[Conic section#Eccentricity, focus and directrix\|directrix]], which is a given line not containing the focus. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity ''e''. If ''e'' is between zero and one the conic is an ellipse; if ''e''=1 the conic is a parabola; and if ''e''>1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix is a [[line at infinity]], so the eccentricity is zero, then the conic is a circle.

			===Defining conics in terms of a focus and a directrix circle===

			It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle. The ellipse thus generated has its second focus at the center of the directrix circle, and the ellipse lies entirely within the circle.

			For the parabola, the center of the directrix moves to the point at infinity (see [[projective geometry]]). The directrix 'circle' becomes a curve with zero curvature, indistinguishable from a straight line. The two arms of the parabola become increasingly parallel as they extend, and 'at infinity' become parallel; using the principles of projective geometry, the two parallels intersect at the point at infinity and the parabola becomes a closed curve (elliptical projection).

			To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distance between the center of this circle and the focus; thus, the focus is outside the directrix circle. The arms of the hyperbola approach asymptotic lines and the 'right-hand' arm of one branch of a hyperbola meets the 'left-hand' arm of the other branch of a hyperbola at the point at infinity; this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity. The two branches of a hyperbola are thus the two (twisted) halves of a curve closed over infinity.

			In projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others.

			===Astronomical significance===
			In the [[gravitation]]al [[two-body problem]], the orbits of the two bodies are described by two overlapping conic sections each with one of their foci being coincident at the [[center of mass]] ([[Barycentric coordinates (astronomy)\|barycenter]]).

			==Cartesian and Cassini ovals==

			A [[Cartesian oval]] is the set of points for each of which the [[weighted sum]] of the distances to two given foci is a constant. If the weights are equal, the special case of an ellipse results.

			A [[Cassini oval]] is the set of points for each of which the product of the distances to two given foci is a constant.

			==Generalization==
			The concept of a focus can be generalized to arbitrary algebraic curves. Let ''C'' be a curve of class ''m'' and let ''I'' and ''J'' denote the [[circular points at infinity]]. Draw the ''m'' tangents to ''C'' through each of ''I'' and ''J''. There are two sets of ''m'' lines which will have ''m''<sup>2</sup> points of intersection, with exceptions in some cases due to singularities, etc. These points of intersection are the defined to be the foci of ''C''. In other words, a point ''P'' is a focus if both ''PI'' and ''PJ'' are tangent to ''C''. When ''C'' is a real curve, only the intersections of conjugate pairs are real, so there are ''m'' in a real foci and ''m''<sup>2</sup>−''m'' imaginary foci. When ''C'' is a conic, the real foci defined this way are exactly the foci which can be used in the geometric construction of ''C''.

			==Confocal curves==
			Let ''P''<sub>1</sub>, ''P''<sub>2</sub>, …, ''P<sub>m</sub>'' be given as foci of a curve ''C'' of class ''m''. Let ''P'' be the product of the tangential equations of these points and ''Q'' the product of the tangential equations of the circular points at infinity. Then all the lines which are common tangents to both ''P''=0 and ''Q''=0 are tangent to ''C''. So, by the [[AF+BG theorem]], the tangential equation of ''C'' has the form ''HP''+''KQ''=0. Since ''C'' has class ''m'', ''H'' must be a constant and ''K'' but have degree less than or equal to ''m''−2. The case ''H''=0 can be eliminated as degenerate, so the tangential equation of ''C'' can be written as ''P''+''fQ''=0 where ''f'' is an arbitrary polynomial of degree ''m''−2.<ref>Follows Hilton p. 69 with an appeal to AF+BG for simplification.</ref>

			For example, let ''P''<sub>1</sub>=(1,0), ''P''<sub>2</sub>=(−1,0). The tangential equations are ''X''+1=0 and ''X''−1=0 so ''P''= ''X''<sup>2</sup>-1=0. The tangential equations for the circular points at infinity are ''X''+''iY''=0 and ''X''−''iY''=0 so ''Q''=''X''<sup>2</sup>+''Y''<sup>2</sup>. Therefore the tangential equation for a conic with the given foci is ''X''<sup>2</sup>-1+''c''(''X''<sup>2</sup>+''Y''<sup>2</sup>)=0, or (1+''c'')''X''<sup>2</sup>+''cY''<sup>2</sup>=1 where ''c'' is an arbitrary constant. In point coordinates this becomes
			:<math>\frac{x^2}{1+c}+\frac{y^2}{c}=1.</math>

			==References==
			{{reflist}}
			*{{cite book \|title=Plane Algebraic Curves\|first=Harold\|last=Hilton\|publisher=Oxford\|year=1920\|page=69
			\|url=http://www.archive.org/details/cu31924001544216}}

			[[Category:Conic sections]]
			[[Category:Geometric centers]]

en>Sun Creator: /* Computer simulation as a building block of a computer experiment */Typo fixing and checking, typos fixed: n.However, → n. However, (2) using AWB (8097)

2012-08-08T14:24:50Z

Computer simulation as a building block of a computer experiment: Typo fixing and checking, typos fixed: n.However, → n. However, (2) using AWB (8097)

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Computer experiment - Revision history

en>IronGargoyle: Reverted edits by 121.54.58.145 (talk) to last revision by 94.79.52.163 (HG)

94.79.52.163: /* Problems with massive sample sizes */

en>Sun Creator: /* Computer simulation as a building block of a computer experiment */Typo fixing and checking, typos fixed: n.However, → n. However, (2) using AWB (8097)