construction of polar coordinates

1 Construction

When a Euclidean plane $E$ is equipped with the usual Cartesiancoordinates, we can represent any point $P$ by a pair of realnumbers $(x,y)$ in a unique fashion. The Cartesian coordinates alsomake $E$ 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 pointare also expressed by a pair of real numbers $(r,\theta )$. But thisis where the similarity ends.

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

To “distinguish” one point from another on the circle, a “secondcoordinate” is needed. To this end, we fix a ray, a ray ofreference, say $x$ and call it the polar axis. With the polaraxis $x$, a point $P_x$ of distance $r$ to $O$ is located. Then anypoint on the circle of radius $r$ can now be located by ameasurement of “how far” it is from $P_x$. This measurementcorresponds to the angle $\theta$ between the polar axis $x$ and theray 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 intothe 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 $P$.

All of the above steps can be carried out in a Euclidean plane,which, in this case, is $E$. Careful readers will, however, see apotential problem in Step 2 above when $P=O$, since one point doesnot determine a unique line segment in $E$. The quick remedy is toset $(r,\theta )=(0,0)$ be the polar coordinates of the polar origin$O$. This is consistent with the way polar coordinates are definedfor $P\ne O$.

The construction establishes a one-to-one correspondence $f$ between$\{(0,0)\}\cup (0,\mathrm{\infty })\times [0,2\pi )$ and $E$. Laterwe will extend $f$ to ${ℝ}^2$ and see that every point $E$has infinitely many representations in polar coordinates, a propertynot shared by the Cartesian coordinates.

Notice also that the choice for measuring angles using thecounterclockwise sweep of the polar axis is arbitrary. We could haveused the clockwise sweep instead. To switch from one choice ofangular measurement to another, we simply perform a reflection$\rho$ about the polar axis (again, this is possible in a Euclideanplane):

We will follow the standard method of measuring angles by using thecounterclockwise sweep of the polar axis described above.

2 Relations with Cartesian coordinates

From the discussion above, we see that $E$ is now equipped with twocoordinate systems. We can now superimpose the two coordinatesystems to seek out any properties between the two systems. First,identify the polar origin with the rectangular origin of theCartesian coordinates (by translation if necessary). Then, line upthe polar axis with the positive ray of the horizontal axis of theCartesian coordinates (by rotation if necessary).

With this identification, the two sets of coordinates of a point $P$can be related by the following equations:

$x=r\cos \theta \mathit{\hspace{1em}\hspace{1em}}\text{ and}\mathit{\hspace{1em}\hspace{1em}}y=r\sin \theta ,$(1)

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

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

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

$$x=r\cos \theta =r\cos (\theta +2n\pi )\mathit{\hspace{1em}}\text{ and}\mathit{\hspace{1em}}y=r\sin \theta =r\sin (\theta +2n\pi ),$$

for all$n\in \mathbb{Z}$. This suggests the identification of$(r,\theta +2n\pi )$ with $(r,\theta )$. The construction so farestablishes a map from $[0,\mathrm{\infty })\times ℝ$ onto $E$, byextending $f$ in the domain of its second polar coordinate.

To complete the rest of the construction, we need to extend thedomain of the first coordinate from the non-negative reals to all of$ℝ$. Again using equations (1), we see that if , then

$$x=r\cos \theta =-r\cos (\theta +\pi )\mathit{\hspace{1em}\hspace{1em}}\text{ and}\mathit{\hspace{1em}\hspace{1em}}y=r\sin \theta =-r\sin (\theta +\pi ).$$

So if we identify$(r,\theta )$ with $(-r,\theta +\pi )$, we have extended $f$ to$\tilde{f}:{ℝ}^2\to E$, completing the construction. Ifthe metric topology is added to $E$, then $\tilde{f}$ is acovering map of $E$.

It’s possible to define additions via polar coordinates. The usualway to go about this is to convert the polar coordinates toCartesian coordinates using equations (1), add, and then convert theresult back to polar coordinates. The formula for additions in polarcoordinates is messy and do not follow any algebraic expressions(involving transcendental functions).

Multiplications by a real scalar can be defined similarly. Thistime, there is a simple formula: $t(r,\theta )=(tr,\theta )$.