HOME

TheInfoList



OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
. Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole.


Properties

Pole and polar have several useful properties: * If a point P lies on the line ''l'', then the pole L of the line ''l'' lies on the polar ''p'' of point P. * If a point P moves along a line ''l'', its polar ''p'' rotates about the pole L of the line ''l''. * If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points. * If a point lies on the conic section, its polar is the tangent through this point to the conic section. * If a point P lies on its own polar line, then P is on the conic section. * Each line has, with respect to a non-degenerated conic section, exactly one pole.


Special case of circles

The pole of a line ''L'' in a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
''C'' is a point Q that is the
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
in ''C'' of the point P on ''L'' that is closest to the center of the circle. Conversely, the polar line (or polar) of a point Q in a circle ''C'' is the line ''L'' such that its closest point P to the center of the circle is the
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
of Q in ''C''. The relationship between poles and polars is reciprocal. Thus, if a point A lies on the polar line ''q'' of a point Q, then the point Q must lie on the polar line ''a'' of the point A. The two polar lines ''a'' and ''q'' need not be parallel. There is another description of the polar line of a point P in the case that it lies outside the circle ''C''. In this case, there are two lines through P which are tangent to the circle, and the polar of P is the line joining the two points of tangency (not shown here). This shows that pole and polar line are concepts in the
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
of the plane and generalize with any nonsingular conic in the place of the circle ''C''.


Polar reciprocation

The concepts of ''a pole is and its polar line'' were advanced in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
. For instance, the polar line can be viewed as the set of projective harmonic conjugates of a given point, the pole, with respect to a conic. The operation of replacing every point by its polar and vice versa is known as a polarity. A polarity is a
correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...
that is also an involution. For some point ''P'' and its polar ''p'', any other point ''Q'' on ''p'' is the pole of a line ''q'' through ''P''. This comprises a reciprocal relationship, and is one in which incidences are preserved.


General conic sections

The concepts of pole, polar and reciprocation can be generalized from circles to other
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s which are the
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
,
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
and
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
. This generalization is possible because conic sections result from a reciprocation of a circle in another circle, and the properties involved, such as incidence and the
cross-ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
, are preserved under all projective transformations.


Calculating the polar of a point

A general
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
may be written as a second-degree equation in the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
(''x'', ''y'') of the plane : A_ x^ + 2 A_ xy + A_ y^ + 2 B_ x + 2 B_ y + C = 0\, where ''A''''xx'', ''A''''xy'', ''A''''yy'', ''B''''x'', ''B''''y'', and ''C'' are the constants defining the equation. For such a conic section, the polar line to a given pole point (ξ, η) is defined by the equation : D x + E y + F = 0\, where ''D'', ''E'' and ''F'' are likewise constants that depend on the pole coordinates (ξ, η) :\begin D &= A_ \xi + A_ \eta + B_ \\ E &= A_ \xi + A_ \eta + B_ \\ F &= B_ \xi + B_ \eta + C\, \end


Calculating the pole of a line

The pole of the line D x + E y + F = 0 , relative to the non-degenerated conic section : A_ x^ + 2 A_ xy + A_ y^ + 2 B_ x + 2 B_ y + C = 0\, can be calculated in two steps. First, calculate the numbers x, y and z from : \begin x \\ y \\ z \end = \begin A_ & A_ & B_ \\ A_ & A_ & B_ \\ B_ & B_ & C \end^ \cdot \begin D \\ E \\ F \end Now, the pole is the point with coordinates \left( \frac , \frac \right)


Tables for pole-polar relations

* Pole-polar relation for an ellipse * Pole-polar relation for a hyperbola * Pole-polar relation for a parabola


Via complete quadrangle

Given four points forming a complete quadrangle, the lines connecting the points cross in an additional three diagonal points. Given a point ''Z'' not on conic ''C'', draw two secants from ''Z'' through ''C'' crossing at points ''A'', ''B'', ''D'', and ''E''. Then these four points form a complete quadrangle with ''Z'' at one of the diagonal points. The line joining the other two diagonal points is the polar of ''Z'', and ''Z'' is the pole of this line.


Applications

Poles and polars were defined by
Joseph Diaz Gergonne Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician. Life In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion becau ...
and play an important role in his solution of the problem of Apollonius. In planar dynamics a pole is a center of rotation, the polar is the force line of action and the conic is the mass–inertia matrix.John Alexiou Thesis, Chapter 5, pp. 80–108
The pole–polar relationship is used to define the
center of percussion The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
of a planar rigid body. If the pole is the hinge point, then the polar is the percussion line of action as described in planar screw theory.


See also

*
Dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Properties Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (ed ...
*
Dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the oth ...
* Polar curve *
Projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
* Projective harmonic conjugates


Bibliography

* * * * The paperback version published by Dover Publications has the . *


References


External links


Interactive animation with multiple poles and polars
at
Cut-the-Knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Interactive animation with one pole and its polar

Interactive 3D with coloured multiple poles/polars - open source
* * * * {{MathWorld, title=Reciprocal curve, urlname=ReciprocalCurve

at Math-abundance Euclidean plane geometry Projective geometry Circles