construction of polar coordinates
When a Euclidean plane is equipped with the usual Cartesian coordinates, we can represent any point by a pair of real numbers in a unique fashion. The Cartesian coordinates also make into a vector space over the reals.
Polar coordinates are an alternative way of describing points in . Like the Cartesian coordinates, polar coordinates of a point are also expressed by a pair of real numbers . But this is where the similarity ends.
To set up the polar coordinates in , we first pick a point of reference and call it the polar origin. For any point in , we can measure the distance between and the polar origin . In fact, along any ray emanating from , we can uniquely identify any point on that ray by its distance from . However, if more than one ray enters the picture, distance alone would not be enough to uniquely identify points in . If we fix the distance , we are looking at a collection of points called a circle, with radius and center .
To “distinguish” one point from another on the circle, a “second coordinate” is needed. To this end, we fix a ray, a ray of reference, say and call it the polar axis. With the polar axis , a point of distance to is located. Then any point on the circle of radius can now be located by a measurement of “how far” it is from . This measurement corresponds to the angle between the polar axis and the ray in question. Furthermore, this angle uniquely identifies a ray. With and , it is now enough to locate any point on uniquely. Operationally, the construction can be broken down into the following sequence of steps:
pick a point and a ray emanating from ,
for any given point , draw a straight line segment connecting and ,
measure the length of the line segment ,
measure the angle between and , by sweeping counterclockwise until it first reaches ,
then are the polar coordinates of .
All of the above steps can be carried out in a Euclidean plane, which, in this case, is . Careful readers will, however, see a potential problem in Step 2 above when , since one point does not determine a unique line segment in . The quick remedy is to set be the polar coordinates of the polar origin . This is consistent with the way polar coordinates are defined for .
The construction establishes a one-to-one correspondence between and . Later we will extend to and see that every point has infinitely many representations in polar coordinates, a property not shared by the Cartesian coordinates.
Notice also that the choice for measuring angles using the counterclockwise sweep of the polar axis is arbitrary. We could have used the clockwise sweep instead. To switch from one choice of angular measurement to another, we simply perform a reflection about the polar axis (again, this is possible in a Euclidean plane):
We will follow the standard method of measuring angles by using the counterclockwise sweep of the polar axis described above.
2 Relations with Cartesian coordinates
From the discussion above, we see that is now equipped with two coordinate systems. We can now superimpose the two coordinate systems to seek out any properties between the two systems. First, identify the polar origin with the rectangular origin of the Cartesian coordinates (by translation if necessary). Then, line up the polar axis with the positive ray of the horizontal axis of the Cartesian coordinates (by rotation if necessary).
With this identification, the two sets of coordinates of a point can be related by the following equations:
where and are respectively the Cartesian and polar coordinates of .
With the pair of equations, we can now show how to extend the one-to-one correspondence (see Section 1) to a map that is -to-one. First, note that if , then no matter what is. Since the Cartesian origin is the polar origin, we identify with the polar origin, for any . This means that the polar origin has uncountably many polar coordinate representations.
Next, if , we see that
for all . This suggests the identification of with . The construction so far establishes a map from onto , by extending in the domain of its second polar coordinate.
To complete the rest of the construction, we need to extend the domain of the first coordinate from the non-negative reals to all of . Again using equations (1), we see that if , then
So if we identify with , we have extended to , completing the construction. If the metric topology is added to , then is a covering map of .
It’s possible to define additions via polar coordinates. The usual way to go about this is to convert the polar coordinates to Cartesian coordinates using equations (1), add, and then convert the result back to polar coordinates. The formula for additions in polar coordinates is messy and do not follow any algebraic expressions (involving transcendental functions).
Multiplications by a real scalar can be defined similarly. This time, there is a simple formula: .
|Title||construction of polar coordinates|
|Date of creation||2013-03-22 15:12:28|
|Last modified on||2013-03-22 15:12:28|
|Last modified by||CWoo (3771)|