SafekipediaPole and polar
Adapted from Wikipedia · Discoverer experience
In geometry, a pole and polar describe a special connection between a point and a line related to a shape called a conic section. When we talk about polar reciprocation, we mean that every point can be turned into a line, and every line can be turned into a point, based on the rules of the conic section. This idea helps us understand how points and lines relate to each other in many math problems. It is a important concept in studying shapes and their properties.
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 that is linked to it through a process called inversion. Similarly, for a point outside the circle, we can draw lines that just touch the circle, and the line connecting where these lines touch the circle is linked to the point.
This matching between points and lines works both ways. If a point lies on a certain line linked to another point, then that other point will lie on a line linked to the first point. These lines may or may not be parallel to each other.
Polar reciprocation
Main article: Correlation (projective geometry)
In geometry, a special idea called polar reciprocation helps us understand how points and lines relate to each other. Imagine you have a point and a line that work together in a unique way. When we talk about a point and its polar line, we mean that each point has a matching line, and each line has a matching point. This matching is called a polarity.
This polarity is a special kind of matching where each point and line swap places while keeping certain rules. For example, if you pick a point and find its matching line, any other point on that line will have a matching line that goes through the first point. This shows how points and lines can depend on each other in a neat, balanced way.
General conic sections
The ideas of a pole and its polar line can be used not just for circles, but also for other shapes called conic sections. These include ellipses, hyperbolas, and parabolas. This works 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:
- Pole-polar relation for an ellipse
- Pole-polar relation for a hyperbola
- Pole-polar relation for a parabola
| conic | equation | polar of point P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})} |
|---|---|---|
| circle | x 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} |
| parabola | y = a x 2 {\displaystyle y=ax^{2}} | y + y 0 = 2 a x 0 x {\displaystyle y+y_{0}=2ax_{0}x} |
| conic | equation | pole of line u x + v y = w |
|---|---|---|
| circle | x 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)} |
| parabola | y = 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
This article is a child-friendly adaptation of the Wikipedia article on Pole and polar, available under CC BY-SA 4.0.