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[parent] construction of polar coordinates (Derivation)

Construction

When a Euclidean plane $ E$ is equipped with the usual Cartesian coordinates, we can represent any point $ P$ by a pair of real numbers $ (x,y)$ in a unique fashion. The Cartesian coordinates also make $ E$ into a vector space over the reals.

Polar coordinates are an alternative way of describing points in $ E$. Like the Cartesian coordinates, polar coordinates of a point are also expressed by a pair of real numbers $ (r,\theta)$. But this is where the similarity ends.

To set up the polar coordinates in $ E$, we first pick a point of reference $ O$ and call it the polar origin. For any point $ P$ in $ E$, we can measure the distance $ r$ between $ P$ and the polar origin $ O$. In fact, along any ray emanating from $ O$, we can uniquely identify any point on that ray by its distance $ r$ from $ O$. However, if more than one ray enters the picture, distance alone would not be enough to uniquely identify points in $ E$. If we fix the distance $ r$, we are looking at a collection of points called a circle, with radius $ r$ and center $ O$.

\includegraphics{polar_1.eps}
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 $ x$ and call it the polar axis. With the polar axis $ x$, a point $ P_x$ of distance $ r$ to $ O$ is located. Then any point on the circle of radius $ r$ can now be located by a measurement of “how far” it is from $ P_x$. This measurement corresponds to the angle $ \theta$ between the polar axis $ x$ and the ray in question. Furthermore, this angle uniquely identifies a ray. With $ O$ and $ x$, it is now enough to locate any point on $ E$ uniquely. Operationally, the construction can be broken down into the following sequence of steps:
  1. pick a point $ O$ and a ray $ x$ emanating from $ O$,
  2. for any given point $ P$, draw a straight line segment connecting $ O$ and $ P$,
  3. measure the length $ r$ of the line segment $ \overline{OP}$,
  4. measure the angle $ \theta$ between $ x$ and $ \overline{OP}$, by sweeping $ x$ counterclockwise until it first reaches $ \overline{OP}$,
  5. then $ (r,\theta)$ are the polar coordinates of $ P$.
\includegraphics{polar_2.eps}
All of the above steps can be carried out in a Euclidean plane, which, in this case, is $ E$. Careful readers will, however, see a potential problem in Step 2 above when $ P=O$, since one point does not determine a unique line segment in $ E$. The quick remedy is to set $ (r,\theta)=(0,0)$ be the polar coordinates of the polar origin $ O$. This is consistent with the way polar coordinates are defined for $ P\neq O$.

The construction establishes a one-to-one correspondence $ f$ between $ \lbrace (0,0)\rbrace \cup (0,\infty)\times[0,2\pi)$ and $ E$. Later we will extend $ f$ to $ \mathbb{R}^2$ and see that every point $ E$ 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 $ \rho$ about the polar axis (again, this is possible in a Euclidean plane):

\includegraphics{polar_3.eps}
We will follow the standard method of measuring angles by using the counterclockwise sweep of the polar axis described above.

Relations with Cartesian coordinates

From the discussion above, we see that $ E$ 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).
\includegraphics{polar_4.eps}
With this identification, the two sets of coordinates of a point $ P$ can be related by the following equations:
$\displaystyle x=r\cos\theta$    and$\displaystyle \qquad y=r\sin\theta,$     (1)

where $ (x,y)$ and $ (r,\theta)$ are respectively the Cartesian and polar coordinates of $ P$.

With the pair of equations, we can now show how to extend the one-to-one correspondence $ f$ (see Section 1) to a map $ \widetilde{f}$ that is $ \infty$-to-one. First, note that if $ r=0$, then $ x=y=0$ no matter what $ \theta$ is. Since the Cartesian origin is the polar origin, we identify $ (0,\theta)$ with the polar origin, for any $ \theta\in\mathbb{R}$. This means that the polar origin has uncountably many polar coordinate representations.

Next, if $ r>0$, we see that

$\displaystyle x=r\cos\theta=r\cos(\theta+2n\pi)$    and$\displaystyle \quad y=r\sin\theta=r\sin(\theta+2n\pi),$
for all $ n\in\mathbb{Z}$. This suggests the identification of $ (r,\theta+2n\pi)$ with $ (r,\theta)$. The construction so far establishes a map from $ [0,\infty)\times\mathbb{R}$ onto $ E$, by extending $ f$ 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 $ \mathbb{R}$. Again using equations (1), we see that if $ r<0$, then

$\displaystyle x=r\cos\theta=-r\cos(\theta+\pi)$    and$\displaystyle \qquad y=r\sin\theta=-r\sin(\theta+\pi).$
So if we identify $ (r,\theta)$ with $ (-r,\theta+\pi)$, we have extended $ f$ to $ \widetilde{f}:\mathbb{R}^2\to E$, completing the construction. If the metric topology is added to $ E$, then $ \widetilde{f}$ is a covering map of $ E$.

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: $ t(r,\theta)=(tr,\theta)$.



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Also defines:  polar origin, polar axis

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Cross-references: simple, scalar, multiplications, transcendental functions, expressions, algebraic, additions, covering map, metric topology, complete, domain, onto, map, section, equations, coordinates, rotation, axis, positive, line, necessary, translation, origin, coordinate systems, reflection, property, representations, one-to-one correspondence, consistent, potential, length, line segment, straight, sequence, angle, center, radius, circle, collection, fix, ray, distance, measure, similarity, polar coordinates, vector space, real numbers, point, represent, Cartesian coordinates, Euclidean plane
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This is version 7 of construction of polar coordinates, born on 2005-04-26, modified 2006-04-17.
Object id is 6968, canonical name is ConstructionOfPolarCoordinates.
Accessed 4607 times total.

Classification:
AMS MSC: 51-01 (Geometry :: Instructional exposition )
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