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Pole and polar

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Pole and polar

In geometry, a pole and polar describe a special link between a point and a line. This link is about a shape called a conic section.

Every point can become a line, and every line can become a point, following the rules of the conic section. This helps us see how points and lines are connected in many math problems. It is an important idea when we study shapes and their features.

Properties

Pole and polar have some interesting relationships:

  • If a point is on a line, then the special line linked to that point (its polar) will pass through the special point linked to the line (its pole).
  • When a point moves along a line, its linked line (polar) moves around the special point (pole) of that line.
  • Each line has exactly one special point (pole) linked to it with respect to a given shape (conic section).

Special case of circles

See also: Inversive geometry

When we look at a circle, there is a special way to match points and lines. For a line close to the circle, we can find a point linked to it. For a point outside the circle, we can draw lines that touch the circle, and the line connecting these points is linked to the point.

This matching between points and lines works both ways. If a point is on a line linked to another point, then that other point will be on a line linked to the first point. These lines may or may not be parallel.

Polar reciprocation

Main article: Correlation (projective geometry)

In geometry, polar reciprocation is an idea that shows how points and lines are connected. Imagine you have a point and a line that work together in a special way. Each point has a matching line, and each line has a matching point. This matching is called polarity.

Polarity is a way that points and lines switch places but still follow certain rules. For example, if you choose a point and find its matching line, any other point on that line will have a matching line that passes through the first point. This shows how points and lines can rely on each other in a balanced way.

General conic sections

The ideas of a pole and its polar line work not just for circles, but also for other shapes called conic sections. These include ellipses, hyperbolas, and parabolas. This is because these shapes can be made by changing a circle in certain ways, and important properties stay the same.

In geometry, any two lines in a flat space will meet at a point. If we have four points that form a special arrangement called a complete quadrangle, the lines connecting these points will cross at three extra points called diagonal points.

If we pick a point Z that is not on a conic shape C, we can draw two lines from Z that cross the shape at points A, B, D, and E. These four points make a complete quadrangle, and Z is one of the diagonal points. The line connecting the other two diagonal points is the polar of Z, and Z is the pole of that line.

Main articles:

conicequationpolar of point P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})}
circlex 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} x 0 x + y 0 y = r 2 {\displaystyle x_{0}x+y_{0}y=r^{2}}
ellipse( x a ) 2 + ( y b ) 2 = 1 {\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1} x 0 x a 2 + y 0 y b 2 = 1 {\displaystyle {\frac {x_{0}x}{a^{2}}}+{\frac {y_{0}y}{b^{2}}}=1}
hyperbola( x a ) 2 − ( y b ) 2 = 1 {\displaystyle \left({\frac {x}{a}}\right)^{2}-\left({\frac {y}{b}}\right)^{2}=1} x 0 x a 2 − y 0 y b 2 = 1 {\displaystyle {\frac {x_{0}x}{a^{2}}}-{\frac {y_{0}y}{b^{2}}}=1}
parabolay = a x 2 {\displaystyle y=ax^{2}} y + y 0 = 2 a x 0 x {\displaystyle y+y_{0}=2ax_{0}x}
conicequationpole of line u x + v y = w
circlex 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} ( r 2 u w , r 2 v w ) {\displaystyle \left({\frac {r^{2}u}{w}},\;{\frac {r^{2}v}{w}}\right)}
ellipse( x a ) 2 + ( y b ) 2 = 1 {\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1} ( a 2 u w , b 2 v w ) {\displaystyle \left({\frac {a^{2}u}{w}},\;{\frac {b^{2}v}{w}}\right)}
hyperbola( x a ) 2 − ( y b ) 2 = 1 {\displaystyle \left({\frac {x}{a}}\right)^{2}-\left({\frac {y}{b}}\right)^{2}=1} ( a 2 u w , − b 2 v w ) {\displaystyle \left({\frac {a^{2}u}{w}},\;-{\frac {b^{2}v}{w}}\right)}
parabolay = a x 2 {\displaystyle y=ax^{2}} ( − u 2 a v , − w v ) {\displaystyle \left(-{\frac {u}{2av}},\;-{\frac {w}{v}}\right)}

Applications

Poles and polars were first described by Joseph Diaz Gergonne and helped him solve an important geometry problem called the problem of Apollonius.

In the study of moving objects, a pole can be a point where something turns, and the polar is the line where a force acts. This idea helps us understand special points in objects, like the center of percussion. If the pole is a fixed point, the polar shows the direction of the force, as explained in screw theory.

Related articles

Geometry
Conic section
Circle
Inversive geometry

This article is a child-friendly adaptation of the Wikipedia article on Pole and polar, available under CC BY-SA 4.0.