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Difference between revisions of "Pole"

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The pole of coordinates is the origin in [[Polar coordinates|polar coordinates]].
 
The pole of coordinates is the origin in [[Polar coordinates|polar coordinates]].
  
 
A pole is the centre of an [[Inversion|inversion]].
 
A pole is the centre of an [[Inversion|inversion]].
  
The pole of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073480/p0734801.png" /> with respect to a [[Conic|conic]] is the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073480/p0734802.png" /> such that the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073480/p0734803.png" /> is the [[Polar|polar]] of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073480/p0734804.png" /> with respect to the conic.
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The pole of the straight line $p$ with respect to a [[Conic|conic]] is the point $P$ such that the line $p$ is the [[Polar|polar]] of the point $P$ with respect to the conic.
  
  
  
 
====Comments====
 
====Comments====
For poles of (analytic) functions see [[Pole (of a function)|Pole (of a function)]]. Sometimes the word  "pole"  is used for the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073480/p0734805.png" /> (North pole) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073480/p0734806.png" /> (South pole) of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073480/p0734807.png" />, with centre at the origin.
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For poles of (analytic) functions see [[Pole (of a function)|Pole (of a function)]]. Sometimes the word  "pole"  is used for the point $(0,0,1)$ (North pole) or $(0,0,-1)$ (South pole) of the unit sphere in $\mathbf R^3$, with centre at the origin.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR></table>

Latest revision as of 19:31, 28 April 2014

The pole of coordinates is the origin in polar coordinates.

A pole is the centre of an inversion.

The pole of the straight line $p$ with respect to a conic is the point $P$ such that the line $p$ is the polar of the point $P$ with respect to the conic.


Comments

For poles of (analytic) functions see Pole (of a function). Sometimes the word "pole" is used for the point $(0,0,1)$ (North pole) or $(0,0,-1)$ (South pole) of the unit sphere in $\mathbf R^3$, with centre at the origin.

References

[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
How to Cite This Entry:
Pole. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pole&oldid=12562
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article