Cartesian coordinates.

*polar coordinates*, because you treat the crossing point of the axes as a pole from which everything radiates out. In the image below, click on the point and drag it around to see how its polar coordinates $(r, \theta)$ change (degrees are measured in radians). Some shapes that are hard to describe in Cartesian coordinates are easier to describe using polar coordinates. For example, think of a circle of radius $2$ centred on the point $(0,0)$. It is made up of all the points that lie a distance $2$ from $(0,0)$. In polar coordinates these are all the points with coordinates $(2,\theta)$, where $\theta$ can take any value at all. In Cartesian coordinates this circle is a little harder to describe. It is made up of all points with coordinates $(x,y)$ where $$x^2+y^2=2^2=4.$$ (This follows from Pythagoras' theorem.)

A circle. It is made up of all points whose Cartesian coordinates (*x*, *y*) satisfy *x*^{2}+*y*^{2}= 4 and whose polar coordinates (*r*, *θ*) satisfy *r*=2. The points shown has Cartesian coordinates (√2, √2) and polar coordinates (2,45), with the angle measured in degrees.

*Archimedean spiral*. The movie below shows the points with coordinates $(r,r)$, as $r$ grows from $0$ to $20\pi$ (which corresponds to ten full turns of $\theta=r$). It's a lot harder to describe such an Archimedean spiral in Cartesian coordinates! Finally, we look at points whose second polar coordinate $r$ is equal to $r=e^{\theta/5}$, where $\theta$ is the second polar coordinate and $e \approx 2.718$ is the base of the natural logarithm. In this case, the first coordinate $r=e^{\theta/5}$ (the distance from the corresponding point to $(0,0)$) grows faster than the second coordinate $\theta$ (the angle). The result is a spiral whose turns aren't as tight as that of an Archimedean spiral — it's an example of a

*logarithmic spiral*. The movie below shows the points with coordinates $(e^{\theta/5},\theta)$, as $\theta$ grows from $0$ to $8\pi$ (corresponding to four full turns).

See *Polar power* for more about Archimedean and logarithmic spirals, as well as other interesting shapes you can draw with polar coordinates.

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## Comments

## Origin polar coordinates

What are the polar coordinates of the origin? You always refer to the origin as (0,0) but these are Cartesian coordinates.

## Origin polar coordinates

I think that (0,0) is perfectly serviceable as the polar coordinates, since we can reason a radius of 0 from the origin, with an angle of 0 radians.

## Origin with an angle

The origin in polar co-ordinates is an improvement on Cartesian coordinates (0, θ) is the origin for any θ. It contains (for continuous curves) information about the

angleat which a curve intersected with the origin, which can actually be an important fact for sketching the curve.