Polar
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) |
Polar. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar&oldid=48225