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==Polar of a point with respect to a conic==
 
The polar of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734001.png" /> with respect to a non-degenerate conic is the line containing all points harmonically conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734002.png" /> with respect to the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734004.png" /> of intersection of the conic with secants through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734005.png" /> (cf. [[Cross ratio|Cross ratio]]). The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734006.png" /> is called the [[Pole|pole]]. If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734007.png" /> lies outside the conic, then the polar passes through the points of contact of the two tangent lines that can be drawn through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734008.png" /> (see Fig. a). If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734009.png" /> lies on the curve, then the polar is the tangent to the curve at this point. If the polar of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340010.png" /> passes through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340011.png" />, then the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340012.png" /> passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340013.png" /> (see Fig. b).
 
The polar of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734001.png" /> with respect to a non-degenerate conic is the line containing all points harmonically conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734002.png" /> with respect to the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734004.png" /> of intersection of the conic with secants through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734005.png" /> (cf. [[Cross ratio|Cross ratio]]). The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734006.png" /> is called the [[Pole|pole]]. If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734007.png" /> lies outside the conic, then the polar passes through the points of contact of the two tangent lines that can be drawn through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734008.png" /> (see Fig. a). If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p0734009.png" /> lies on the curve, then the polar is the tangent to the curve at this point. If the polar of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340010.png" /> passes through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340011.png" />, then the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340012.png" /> passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340013.png" /> (see Fig. b).
  
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<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><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Busemann,  P.J. Kelly,  "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Coolidge,  "Algebraic plane curves" , Dover, reprint  (1959)  pp. 195</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><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Busemann,  P.J. Kelly,  "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Coolidge,  "Algebraic plane curves" , Dover, reprint  (1959)  pp. 195</TD></TR></table>
  
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==Polar of a subset of a topological vector space==
 
The polar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340020.png" /> of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340021.png" /> in a locally convex topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340022.png" /> is the set of functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340023.png" /> in the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340024.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340026.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340027.png" /> is the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340028.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340029.png" />). The bipolar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340030.png" /> is the set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340031.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340032.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340034.png" />.
 
The polar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340020.png" /> of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340021.png" /> in a locally convex topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340022.png" /> is the set of functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340023.png" /> in the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340024.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340026.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340027.png" /> is the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340028.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340029.png" />). The bipolar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340030.png" /> is the set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340031.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340032.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073400/p07340034.png" />.
  

Revision as of 20:42, 18 April 2020

Polar of a point with respect to a conic

The polar of a point with respect to a non-degenerate conic is the line containing all points harmonically conjugate to with respect to the points and of intersection of the conic with secants through (cf. Cross ratio). The point is called the pole. If the point lies outside the conic, then the polar passes through the points of contact of the two tangent lines that can be drawn through (see Fig. a). If the point lies on the curve, then the polar is the tangent to the curve at this point. If the polar of the point passes through a point , then the polar of passes through (see Fig. b).

Figure: p073400a

Figure: p073400b

Every non-degenerate conic determines a bijection between the set of points of the projective plane and the set of its straight lines, which is a polarity (a polar transformation). Figures that correspond under this transformation are called mutually polar. A figure coinciding with its polar figure is called autopolar, or self-polar (see, for example, the self-polar triangle in Fig. b).

Analogously one defines the polar (polar plane) of a point with respect to a non-degenerate surface of the second order.

The concept of a polar relative to a conic can be generalized to curves of order . Here, a given point of the plane is put into correspondence with polars with respect to the curve. The first of these polars is a curve of order , the second, which is the polar of the given point relative to the first polar, has order , etc., and, finally, the -st polar is a straight line.

References

[1] N.V. Efimov, "Higher geometry" , MIR (1980) (Translated from Russian)
[2] M.M. Postnikov, "Analytic geometry" , Moscow (1973) (In Russian)


Comments

References

[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a3] H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)
[a4] J.L. Coolidge, "Algebraic plane curves" , Dover, reprint (1959) pp. 195

Polar of a subset of a topological vector space

The polar of a subset in a locally convex topological vector space is the set of functionals in the dual space for which for all (here is the value of at ). The bipolar is the set of vectors in the space for which for all .

The polar is convex, balanced and closed in the weak- topology . The bipolar is the weak closure of the convex balanced hull of the set . In addition, . If is a neighbourhood of zero in the space , then its polar is a compactum in the weak- topology (the Banach–Alaoglu theorem).

The polar of the union of any family of sets in is the intersection of the polars of these sets. The polar of the intersection of weakly-closed convex balanced sets is the closure in the weak- topology of the convex hull of their polars. If is a subspace of , then its polar coincides with the subspace of orthogonal to .

As a fundamental system of neighbourhoods of zero defining the weak- topology of the space one can take the system of sets of the form where runs through all finite subsets of .

A subset of functionals of the space is equicontinuous if and only if it is contained in the polar of some neighbourhood of zero.

References

[1] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)

V.I. Lomonosov

Comments

References

[a1] G. Köthe, "Topological vector spaces" , 1 , Springer (1979) (Translated from German)
[a2] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
[a3] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
How to Cite This Entry:
Polar. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar&oldid=45417
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article