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| {{Redirect|Polar line|the railway line|Polar Line}}
| | It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. s and intelligently including a substantial amount of key words in the title tags, image links, etc. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website. <br><br>As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. The higher your blog ranks on search engines, the more likely people will find your online marketing site. We also help to integrate various plug-ins to expand the functionalities of the web application. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. Enough automated blog posts plus a system keeps you and your clients happy. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>Word - Press has plenty of SEO benefits over Joomla and Drupal. In case you have any queries regarding wherever along with the best way to utilize [http://rlpr.co/wordpress_backup_plugin_291065 wordpress dropbox backup], you possibly can email us at our web page. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. The company gains commission from the customers' payment. See a product, take a picture, and it gives you an Amazon price for that product, or related products. <br><br>This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. In fact portfolio Word - Press themes is a smooth and attractive but considerably flawed Word - Press theme in creating simpler to the photographers or designers to develop a specific internet site showcasing their most current perform since it appear modern-day and has fantastic typography and large photographs which would develop an attractive wanting portfolio internet site. While deciding couple should consider the expertise of the doctor,clinics success rate,the costs of fertility treatment,including fertility tests and IVF costs and overall ones own financial budget. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account. |
| [[File:Pole and polar.svg|thumb|right|The polar line ''q'' to a point '''Q''' with respect to a circle of radius ''r'' centered on the point '''O'''. The point '''P''' is the [[inversive geometry#Circle inversion|inversion point]] of '''Q'''; the polar is the line through '''P''' that is perpendicular to the line containing '''O''', '''P''' and '''Q'''.]]
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| In [[geometry]], the terms '''pole''' and '''polar''' are used to describe a point and a line that have a unique reciprocal relationship with respect to a given [[conic section]]. If the point lies on the conic section, its polar is the tangent line to the conic section at that point.
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| For a given circle, '''reciprocation''' in a circle means to transform each point in the plane into its polar line and each line in the plane into its pole.
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| ==Special case of circles==
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| The pole of a line ''L'' in a [[circle]] ''C'' is a point '''P''' that is the [[inversive geometry#Circle inversion|inversion]] in ''C'' of the point '''Q''' on ''L'' that is closest to the center of the circle. Conversely, the '''polar line''' (or '''polar''') of a point '''P''' in a circle ''C'' is the line ''L'' such that its closest point '''Q''' to the circle is the [[inversion (geometry)#Circle inversion|inversion]] of '''P''' in ''C''.
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| [[File:Conjugate points and lines.svg|thumb|left|If a point '''A''' lies on the polar line ''q'' of another point '''Q''', then '''Q''' lies on the polar line ''a'' of '''A'''. More generally, the polars of all the points on the line ''q'' must pass through its pole '''Q'''.]]
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| The relationship between poles and polars is reciprocal. Thus, if a point '''Q''' is on the polar line ''A'' of a point '''P''', then the point '''P''' must lie on the polar line ''B'' of the point '''Q'''. The two polar lines ''A'' and ''B'' need not be parallel.
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| There is another description of the polar line of a point '''P''' in the case that it lies outside the circle ''C''. In this case, there are two lines through '''P''' which are [[tangent lines to circles|tangent to the circle]], and the polar of '''P''' is the line joining the two points of tangency (not shown here). This shows that [[projective harmonic conjugates#Projective conics|pole and polar line]] are concepts in the [[projective geometry]] of the [[projective plane|plane]] and generalize with any [[conic|nonsingular conic]] in the place of the circle ''C''.
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| ==Reciprocation and projective duality==
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| [[File:Incidence poles polars.svg|thumb|right|Illustration of the duality between points and lines, and the double meaning of "incidence". If two lines ''a'' and ''k'' pass through a single point '''Q''', then the polar ''q'' of '''Q''' joins the poles '''A''' and '''K''' of the lines ''a'' and ''k'', respectively.]]
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| The concepts of ''a pole and its polar line'' were advanced in [[projective geometry]]. For instance, the polar line can be viewed as the set of [[projective harmonic conjugates]] of a given point, the pole, with respect to a conic. The operation of replacing every point by its polar and vice versa is sometimes known as '''reciprocation.'''
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| ==General conic sections==
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| [[File:Pool poollijn.svg|thumb|Line ''p'' is the polar line to point '''P''', ''l'' to '''L''' and ''m'' to '''M''']]
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| [[File:Pool_poollijn_eig.svg|thumb| ''p'' is the polar line to point '''P''' ; ''m'' is the polar line to '''M''']]
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| The concepts of pole, polar and reciprocation can be generalized from circles to other [[conic section]]s which are the [[ellipse]], [[hyperbola]] and [[parabola]]. This generalization is possible because conic sections result from a reciprocation of a circle in another circle, and the properties involved, such as [[incidence (geometry)|incidence]] and the [[cross-ratio]], are preserved under all [[projective transformation]]s.
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| === Calculating the polar of a point ===
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| A general [[conic section]] may be written as a second-degree equation in the [[Cartesian coordinates]] (''x'', ''y'') of the [[plane (geometry)|plane]]
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| :<math>
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| A_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0\,
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| </math> | |
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| where ''A''<sub>''xx''</sub>, ''A''<sub>''xy''</sub>, ''A''<sub>''yy''</sub>, ''B''<sub>''x''</sub>, ''B''<sub>''y''</sub>, and ''C'' are the constants defining the equation. For such a conic section, the polar line to a given pole point (ξ, η) is defined by the equation
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| :<math> | |
| D x + E y + F = 0\,
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| </math>
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| where ''D'', ''E'' and ''F'' are likewise constants that depend on the pole coordinates (ξ, η)
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| :<math>
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| D = A_{xx} \xi + A_{xy} \eta + B_{x}\,
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| </math>
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| :<math>
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| E = A_{xy} \xi + A_{yy} \eta + B_{y}\,
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| </math>
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| :<math>
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| F = B_{x} \xi + B_{y} \eta + C\,
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| </math>
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| === Calculating the pole of a line ===
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| The pole of the line <math> D x + E y + F =0 </math>, relative to the non-degenerated conic section | |
| :<math>
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| A_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0\,
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| </math>
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| can be calculated in two steps.
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| First, calculate the numbers x, y and z from
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| :<math> \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} A_{xx} & A_{xy} & B_{x} \\ A_{xy} & A_{yy} & B_{y}\\ B_{x} & B_{y} & C \end{bmatrix}^{-1} \cdot \begin{bmatrix} D \\ E \\ F \end{bmatrix} </math>
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| Now, the pole is the point with coordinates <math> ( \frac{x}{z} , \frac{y}{z} ) </math>
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| ==Properties==
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| Poles and polars have several useful properties.
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| If a point '''P''' lies on a line ''l'', then the pole '''L''' of the line ''l'' lies on the polar ''p'' of point '''P'''.
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| If a point '''P''' moves along a line ''l'', its polar ''p'' rotates about the pole '''L''' of the line ''l''.
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| If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points.
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| If a point lies on the conic section, its polar is the tangent through this point to the conic section.
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| If a point '''P''' lies on its own polar line, then '''P''' is on the conic section.
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| Each line has, with respect to a non-degenerated conic section, exactly one pole.
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| ==Applications==
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| Poles and polars were defined by [[Joseph Diaz Gergonne]] and play an important role in his solution of the [[problem of Apollonius]].<ref>{{cite web | url=http://www.ajur.uni.edu/v3n1/Gisch%20and%20Ribando.pdf | title=Apollonius’ Problem: A Study of Solutions and Their Connections | accessdate=2013-06-04}}</ref>
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| In planar dynamics a pole is a center of rotation, the polar is the force line of action and the conic is the mass-inertia matrix.<ref>[http://helix.gatech.edu/Papers/1999/AlexiouThesis.pdf John Alexiou Thesis, Chapter 5, pp. 80–108]</ref> The pole-polar relationship is used to define the [[center of percussion]] of a planar rigid body. If the pole is the hinge point, then the polar is the percussion line of action as described in planar [[screw theory]]. | |
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| ==See also==
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| * [[Dual polygon]]
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| * [[Dual polyhedron]]
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| * [[Projective geometry]]
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| * [[Projective harmonic conjugates]]
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| * [[Polar curve]]
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| * [[Pascal line formula]]
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| ==Bibliography==
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| * {{cite book | author = Johnson RA | year = 1960 | title = Advanced Euclidean Geometry: An Elementary treatise on the geometry of the Triangle and the Circle | publisher = Dover Publications | location = New York | pages = 100–105}}
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| * {{cite book | title = Geometry Revisited | author = [[Harold Scott MacDonald Coxeter|Coxeter HSM]], [[S. L. Greitzer|Greitzer SL]]| year = 1967 | publisher = [[Mathematical Association of America|MAA]] | location = [[Washington, D.C.|Washington]] | isbn = 978-0-88385-619-2 | pages = 132–136, 150}}
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| * {{cite book | author = Gray J J | year = 2007 | title = Worlds Out of Nothing: A Course in the history of Geometry in the 19th century | publisher = Springer Verlag | location = London | isbn = 978-1-84628-632-2 | pages = 21}}
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| * {{cite book | author = Korn GA, Korn TM | year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | pages = pp43–45 | lccn = 5914456}} The paperback version published by Dover Publications has the ISBN 978-0-486-41147-7.
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| * {{cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 190–191}}
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/PolePolar.shtml Interactive animation with multiple poles and polars] at [[Cut-the-Knot]]
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| * [http://www.geogebra.org/de/examples/polare/polare_kreis.html Interactive animation with one pole and its polar]
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| * {{MathWorld|title=Polar|urlname=Polar}}
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| * {{MathWorld|title=Reciprocation|urlname=Reciprocation}}
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| * {{MathWorld|title=Inversion pole|urlname=InversionPole}}
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| * {{MathWorld|title=Reciprocal curve|urlname=ReciprocalCurve}}
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| * [http://www.ping.be/~ping1339/pole.htm Tutorial] at Math-abundance
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| [[Category:Euclidean plane geometry]]
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| [[Category:Projective geometry]]
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| [[Category:Circles]]
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It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. s and intelligently including a substantial amount of key words in the title tags, image links, etc. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website.
As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. The higher your blog ranks on search engines, the more likely people will find your online marketing site. We also help to integrate various plug-ins to expand the functionalities of the web application. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. Many times the camera is following Mia, taking in her point of view in almost every frame.
You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. Enough automated blog posts plus a system keeps you and your clients happy. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog.
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