From Wikipedia, the free encyclopedia
A=Geographic poles, B=Geomagnetic poles, C=Magnetic poles
A geographical pole is either of two points on the surface of a rotating . It is the place where the
meets the surface of the planet. The
of a body is 90
north of the . The
lies 90 degrees south of the equator.
Wikimedia Commons has media related to .
can be made longer. You can help Wikipedia by .
: Hidden category:Poles and Polars
Poles and Polars
Poles and polars come in pairs. Po polars are straight lines in the same plane. The correspondence between points and lines that makes them into poles and polars , including a pair of straight lines, and, more generally, for algebraic curves. Below, I define the pole/polar correspondence with respect to a circle.
So let there be a circle w with center O and radius R. Let A be an arbitrary point, except O, whose
in w is A'. OA&OA' = R2. A' is collinear with O and A. Let a be the line through A' perpendicular to OA' (or OA, which is the same.) Then a is called the polar of A, while A is the pole of a, w.r.t. to w.
If A is outside w, we may draw two tangents from A to w. Then the polar a is the line through the two points of tangency. (It's this property that could be used as a definition of polar with respect to an algebraic curve.) If A lies on w, its polar a is just the tangent to w at A.
This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.
The applet allows adding any number of pole/polar pairs. This can be done in two ways. You can add a pole which will appear as a small hollow circle that could be dragged. Its polar will be drawn automatically. It is also possible to add a polar - a straight (draggable) line defined by two draggable points. Its polar will then appear as a small filled circle.
Observe the main properties of poles and their polars:
The polar of a pole that lies inside the circle of reference lies in its entirety outsdie the circle.
The polar of a pole on the circle is tangent to the circle at the pole.
The polar of a pole that lies outside the circle of reference crosses the circle at two points. The lines joining the points to the pole are .
If point A lies on the polar of point B, then point B lies on the polar of point A. (.)
The pole of a line through two poles lies at the intersection of their polars.
The polar of a point which is the intersection of two polars, is the line through the corresponding poles.
The polars of three collinear points are concurrent.
The poles of three concurrent lines are collinear.
The latter properties are a direct consequence of La Hire's theorem. For example, assume points A, B, C with polars a, b, and c, respectively, lie on a straight line m. Let M be the pole of m. Since each of A, B, C lies on m, each of a, b, c passes through M. They are therefore concurrent. Conversely, let three lines a, b, c concur at point M. Then each of A, B, C lies on the polar m of M. They are therefore collinear.
Note that one needs only a straightedge to
of a point. It then follows from 3. that tangents to a circle from a point are straightedge constructible even in the absence of the center of the circle. (Compare that to .)
Let's prove 3:
Let A' be the inverse image of point A lying outside the circle of inversion. A' then lies inside the circle and, by definition,
(1)OA&OA' = R2.
Find points S and T on the circle such that ST is perpendicular to OA at A'. From (1) and the fact that they share an angle at O, triangles OAT and OTA' are similar (SAS). Since angle OA'T is right, so is angle OTA. In other words, AT is perpendicular to the radius OT and is therefore tangent to the circle at T.
Poles and Polars
Poles and Polars
Copyright &
Search by google:请选择错误类型（可多选）：
其他错误（选填）：Antarctica 2008
One World Expedition
In March 2014, Eric Larsen and Ryan Waters, will traverse the Arctic Ocean from Northern Ellesmere Island to the geographic North Pole. The team hopes to cover the nearly 500-mile distance in less than 49 in doing so, they would break the current 'unsupported' expedition speed record set by a Norwegian team in 2006.
Attempted bicycle expedition to the Geographic South Pole. The goal of the expedition was to bicycle form Hercules Inlet to the Geographic South Pole a distance of 730 miles. While Eric was not able to complete the full traverse, he did cover a quarter of the distance to the pole.
In 2009, Eric Larsen began an unprecedented journey to the top, bottom and roof of the world. During a continuous 365-day period, Larsen mounted major expeditions to the North and South Poles and the summit of Mt. Everest. Larsen is the only person to have completed this feat in a one-year time span. To date, less than 20 people in history have completed full expeditions to all three 'poles'.
South Pole: Guiding for Antarctic Logistics and Expeditions (ALE), Eric led two clients to the South Pole via Hercules Inlet. Starting from the edge of the Antarctic continent on the Filchner Ice Shelf, our team skied 730 miles (934 km) to the Geographic South Pole covering the entire distance in 48 days arriving on January 2, 2010.
North Pole: Traveling with Canadian team mate Darcy St. Laurent and English team member Antony Jinman, the team skied, snowshoed and at times even swam roughly 550 miles from Cape Discovery, Ellesmere Island to the Geographic North Pole in 51 days arriving on April 22, 2010 .
Mt. Everest: Climbing with a small Sherpa team, Eric summited Mt. Everest on October 15th, 2010 and stands as the only Fall summit since.
In July of 2009, Ryan Waters, Mark Sheen and Eric traveled to Talkeetna, Alaska to climb Mt. McKinley (Denali) North America's highest peak. Concerned about gaping crevasses common in late season climbing, a large snowfall prior to our departure to base camp substantially improved snow surface. Once on the Kahiltna Glacier, the team was able to make steady progress to 14,000' camp. After a rest/acclimatization day, the team climbed up the headwall to lay a cache at 17,200' camp. Noting the calm conditions, they decided to take advantage of the great weather and reach the summit.
Guiding for Antarctic Logistics and Expeditions (ALE), Eric led a diverse group of four clients to the South Pole via the route originally pioneered by famed mountaineer, Rheinhold Messner. Starting from the edge of the Antarctic continent on the Filchner Ice Shelf, our team skied 580 miles (934 km) to the Geographic South Pole covering the entire distance in 43 days arriving on January 3, 2009.
To date the only successful 'summer' style North Pole expedition. In early May 2006 , Eric and team mate Lonnie Dupre departed Cape Discovery, Ellesmere Island on what was to be one of the most physically demanding and mentally challenging expeditions of his life. Traditionally, Arctic Ocean expeditions had been launched during the coldest time of year while the ice is still thick and stable. Yet freeze-thaw conditions, dangerous shifting ice floes, slush and melt water pools had prevented even the most experienced explorers from even attempting a summer North Pole Expedition. The team spent nearly four years planning and preparing and even developed specially modified canoes that could be pulled like sleds and paddled like boats. After a failed attempt in 2005 from Siberia, he pair achieved the North Pole on July 2nd, after 62 days on the ice. Initially, they had planned to return to land, but due to Lonnie's increasingly worrisome medical condition, they rendezvoused with a Russian Ice Breaker at the North Pole.}