Polar

Polar of a point with respect to a conic

The polar of a point $ P $ with respect to a non-degenerate conic is the line containing all points harmonically conjugate to $ P $ with respect to the points $ M _ {1} $ and $ M _ {2} $ of intersection of the conic with secants through $ P $( cf. Cross ratio). The point $ P $ is called the pole. If the point $ P $ lies outside the conic, then the polar passes through the points of contact of the two tangent lines that can be drawn through $ P $( see Fig. a). If the point $ P $ lies on the curve, then the polar is the tangent to the curve at this point. If the polar of the point $ P $ passes through a point $ Q $, then the polar of $ Q $ passes through $ P $( 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 $ PQR $ 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 $ n $. Here, a given point of the plane is put into correspondence with $ n - 1 $ polars with respect to the curve. The first of these polars is a curve of order $ n - 1 $, the second, which is the polar of the given point relative to the first polar, has order $ n - 2 $, etc., and, finally, the $ ( n - 1 ) $- 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)

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 $ A ^ {o} $ of a subset $ A $ in a locally convex topological vector space $ E $ is the set of functionals $ f $ in the dual space $ E ^ \prime $ for which $ | \langle x , f \rangle | \leq 1 $ for all $ x \in A $( here $ \langle x , f \rangle $ is the value of $ f $ at $ x $). The bipolar $ A ^ {oo} $ is the set of vectors $ x $ in the space $ E $ for which $ | \langle x , f \rangle | \leq 1 $ for all $ f \in A ^ {o} $.

The polar is convex, balanced and closed in the weak- $ * $ topology $ \sigma ( E ^ \prime , E) $. The bipolar $ A ^ {oo} $ is the weak closure of the convex balanced hull of the set $ A $. In addition, $ ( A ^ {oo} ) ^ {o} = A ^ {o} $. If $ A $ is a neighbourhood of zero in the space $ E $, then its polar $ A ^ {o} $ is a compactum in the weak- $ * $ topology (the Banach–Alaoglu theorem).

The polar of the union $ \cup _ \alpha A _ \alpha $ of any family $ \{ A _ \alpha \} $ of sets in $ E $ is the intersection of the polars of these sets. The polar of the intersection of weakly-closed convex balanced sets $ A _ \alpha $ is the closure in the weak- $ * $ topology of the convex hull of their polars. If $ A $ is a subspace of $ E $, then its polar coincides with the subspace of $ E ^ \prime $ orthogonal to $ A $.

As a fundamental system of neighbourhoods of zero defining the weak- $ * $ topology of the space $ E ^ \prime $ one can take the system of sets of the form $ M ^ {o} $ where $ M $ runs through all finite subsets of $ E $.

A subset of functionals of the space $ E ^ \prime $ 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

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)