Polar

Polar of a point with respect to a conic[edit]

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.

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Polar of a subset of a topological vector space[edit]

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 $|⟨x,f⟩|\le 1$ for all $x\in A$( here $⟨x,f⟩$ is the value of $f$ at $x$). The bipolar $A^{oo}$ is the set of vectors $x$ in the space $E$ for which $|⟨x,f⟩|\le 1$ for all $f\in A^o$.

The polar is convex, balanced and closed in the weak- $\ast$ 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- $\ast$ 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- $\ast$ 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- $\ast$ 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.

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V.I. Lomonosov

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