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{{Redirect|Polar line|the railway line|Polar Line}} | |||
[[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'''.]] | |||
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. | |||
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. | |||
==Special case of circles== | |||
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''. | |||
[[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'''.]] | |||
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. | |||
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''. | |||
==Reciprocation and projective duality== | |||
[[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.]] | |||
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.''' | |||
==General conic sections== | |||
[[File:Pool poollijn.svg|thumb|Line ''p'' is the polar line to point '''P''', ''l'' to '''L''' and ''m'' to '''M''']] | |||
[[File:Pool_poollijn_eig.svg|thumb| ''p'' is the polar line to point '''P''' ; ''m'' is the polar line to '''M''']] | |||
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. | |||
=== Calculating the polar of a point === | |||
A general [[conic section]] may be written as a second-degree equation in the [[Cartesian coordinates]] (''x'', ''y'') of the [[plane (geometry)|plane]] | |||
:<math> | |||
A_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0\, | |||
</math> | |||
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 | |||
:<math> | |||
D x + E y + F = 0\, | |||
</math> | |||
where ''D'', ''E'' and ''F'' are likewise constants that depend on the pole coordinates (ξ, η) | |||
:<math> | |||
D = A_{xx} \xi + A_{xy} \eta + B_{x}\, | |||
</math> | |||
:<math> | |||
E = A_{xy} \xi + A_{yy} \eta + B_{y}\, | |||
</math> | |||
:<math> | |||
F = B_{x} \xi + B_{y} \eta + C\, | |||
</math> | |||
=== Calculating the pole of a line === | |||
The pole of the line <math> D x + E y + F =0 </math>, relative to the non-degenerated conic section | |||
:<math> | |||
A_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0\, | |||
</math> | |||
can be calculated in two steps. | |||
First, calculate the numbers x, y and z from | |||
:<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> | |||
Now, the pole is the point with coordinates <math> ( \frac{x}{z} , \frac{y}{z} ) </math> | |||
==Properties== | |||
Poles and polars have several useful properties. | |||
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'''. | |||
If a point '''P''' moves along a line ''l'', its polar ''p'' rotates about the pole '''L''' of the line ''l''. | |||
If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points. | |||
If a point lies on the conic section, its polar is the tangent through this point to the conic section. | |||
If a point '''P''' lies on its own polar line, then '''P''' is on the conic section. | |||
Each line has, with respect to a non-degenerated conic section, exactly one pole. | |||
==Applications== | |||
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> | |||
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]]. | |||
==See also== | |||
* [[Dual polygon]] | |||
* [[Dual polyhedron]] | |||
* [[Projective geometry]] | |||
* [[Projective harmonic conjugates]] | |||
* [[Polar curve]] | |||
* [[Pascal line formula]] | |||
==Bibliography== | |||
* {{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}} | |||
* {{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}} | |||
* {{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}} | |||
* {{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. | |||
* {{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}} | |||
==References== | |||
{{Reflist}} | |||
==External links== | |||
* [http://www.cut-the-knot.org/Curriculum/Geometry/PolePolar.shtml Interactive animation with multiple poles and polars] at [[Cut-the-Knot]] | |||
* [http://www.geogebra.org/de/examples/polare/polare_kreis.html Interactive animation with one pole and its polar] | |||
* {{MathWorld|title=Polar|urlname=Polar}} | |||
* {{MathWorld|title=Reciprocation|urlname=Reciprocation}} | |||
* {{MathWorld|title=Inversion pole|urlname=InversionPole}} | |||
* {{MathWorld|title=Reciprocal curve|urlname=ReciprocalCurve}} | |||
* [http://www.ping.be/~ping1339/pole.htm Tutorial] at Math-abundance | |||
[[Category:Euclidean plane geometry]] | |||
[[Category:Projective geometry]] | |||
[[Category:Circles]] | |||
Revision as of 21:58, 5 December 2012
Name: Jodi Junker
My age: 32
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Home town: Oudkarspel
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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.
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.
Special case of circles
The pole of a line L in a circle C is a point P that is the 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 of P in C.
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.
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 to the circle, and the polar of P is the line joining the two points of tangency (not shown here). This shows that pole and polar line are concepts in the projective geometry of the plane and generalize with any nonsingular conic in the place of the circle C.
Reciprocation and projective duality
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.
General conic sections
The concepts of pole, polar and reciprocation can be generalized from circles to other conic sections 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 and the cross-ratio, are preserved under all projective transformations.
Calculating the polar of a point
A general conic section may be written as a second-degree equation in the Cartesian coordinates (x, y) of the plane
where Axx, Axy, Ayy, Bx, By, 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
where D, E and F are likewise constants that depend on the pole coordinates (ξ, η)
Calculating the pole of a line
The pole of the line , relative to the non-degenerated conic section
can be calculated in two steps.
First, calculate the numbers x, y and z from
Now, the pole is the point with coordinates
Properties
Poles and polars have several useful properties.
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.
If a point P moves along a line l, its polar p rotates about the pole L of the line l.
If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points.
If a point lies on the conic section, its polar is the tangent through this point to the conic section.
If a point P lies on its own polar line, then P is on the conic section.
Each line has, with respect to a non-degenerated conic section, exactly one pole.
Applications
Poles and polars were defined by Joseph Diaz Gergonne and play an important role in his solution of the problem of Apollonius.[1]
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.[2] 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.
See also
- Dual polygon
- Dual polyhedron
- Projective geometry
- Projective harmonic conjugates
- Polar curve
- Pascal line formula
Bibliography
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 The paperback version published by Dover Publications has the ISBN 978-0-486-41147-7.
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
External links
- Interactive animation with multiple poles and polars at Cut-the-Knot
- Interactive animation with one pole and its polar
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I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.- Tutorial at Math-abundance