# 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