Most of us are familiar with the Cartesian coordinate system which assigns to each point in the plane two coordinates . To find you start at the point and walk a distance along the horizontal axis and a distance along the vertical axis (see image on the right).
But there’s another way of locating points on the plane, which is very nice too. To each point assign the pair of numbers , where is the distance from to along a straight radial line, and is the angle formed by that radial line and the positive -axis, measured anti-clockwise from the -axis to the line. These new coordinates are called 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 change (degrees are measured in radians).
How do you express simple shapes in polar coordinates? As you can see from the interactivity above, a ray that starts at the point is given by a particular value of the angle For example, the positive -axis is described by the equation
and the ray that halves the angle between the positive and -axes is given by the equation
Generally, the equation
describes the ray that starts at and makes an angle with the positive -axis.
What about circles? A circle centred on of radius consists of all points that lie at distance from . So we can describe the circle in polar coordinates using the equation
This expression is a lot simpler than the equation of a circle in Cartesian coordinates, which is
The polar equations describing lines that don't pass through the point (0,0) and circles that are not centred on (0,0) are more complicated than their Cartesian versions (see here). But there are shapes whose description using polar coordinates are a lot simpler than their Cartesian counterparts. Here are three of our favourite examples.
Let’s look at all the points whose polar coordinates satisfy the equation
in other words, we are looking at points with polar coordinates Where are they?
The animation below shows the ray corresponding to the angle as ranges from 0 to . The point marked on the ray is the one with coordinates As increases the point moves further away from (Click on the play icon in the bottom left hand corner or use the slider to vary .)
We have the beginning of a spiral!
But why stop at ? You could keep turning the radial line by more than one full turn, through one-and-a-half turns (), two turns (), and so on, round and round. Then, as goes through another full turn from to , the distance to of the point with coordinates increases further, from to . Letting run from to gives us another full turn with points moving out even further. The animation below shows the spiral you get from letting vary from 0 to (The angle varies in little increments, that’s why you see dots rather than a continuous line.)
Letting increase all the way to infinity gives a spiral that loops around an infinite number of times:
This beautiful shape is called an Archimedean spiral, named after the great Greek mathematician Archimedes who discovered it in the third century BC. As you can see from the picture, the loops of the spiral are evenly spaced: if you draw a radial line from , then the distance between two successive intersection points with the spiral is always .
There are other Archimedean spirals too, all characterised by the fact that a radial line intersects the spiral at equal intervals. They can be characterised by the equation
for a positive real number. By playing around with different values for you can convince yourself that controls how tightly the spiral is wound up, and therefore the distance between successive intersection points along a radial line. If you prefer a physical interpretation, an Archimedean spiral is what you get when you trace the path of a point that moves out from the centre at constant speed along a line that rotates with constant angular velocity.
Now let’s look at all points whose polar coordinates satisfy the equation
where is the base of the natural logarithm, .
When , we get
So our shape contains the point with polar coordinates (whose Cartesian coordinates happen to also be ). The animation below shows the radial line corresponding to the angle as it varies from to The point with coordinates is marked too. Again we see the beginning of a spiral but this time a different one.
Again we can let range all the way to infinity, giving a spiral that winds around an infinite number of times. This time, however, the loops of the spiral are not evenly spaced.
This is an example of a logarithmic spiral. It carries this name because instead of writing
you could write
where is the natural logarithm to base .
(There is also a more general form of the logarithmic spiral, given by the equation
where and are positive real constants. )
But there is another trick we can play here: we can allow the angle to become negative! To find a point whose second polar coordinate is negative you measure the angle from the positive -axis in the other direction: clockwise. For example, a point with polar coordinates lies on the negative part of the -axis.
What does this mean for our logarithmic spiral? As moves from towards negative infinity, the radial line given by turns clockwise through one, two, three, and any number of turns. The points with coordinates
lie on that radial line. But now as decreases toward negative infinity, the points move inward, towards This is because
so if is very negative, then is a very large number, so is positive and very close to 0.
The animation below shows the point as varies from down to
Letting range all the way from negative to positive infinity creates a spiral that is infinite in both directions, it has no beginning and no end.
Figure 3: A full logarithmic spiral.
But hang on a second. This picture looks pretty much the same as the one in figure 2 above, even though the x and y-axes here cover a much smaller range, as you can see from the labels! This illustrates a very interesting property of logarithmic spirals. If you zoom in or out of a picture of a logarithmic spiral, the picture you see looks exactly as it did before you zoomed. That feature is called self-similarity. It may be the reason why logarithmic spirals are so common in nature. You can see them in the turns of a snail shell, in many plants and even in the arms of spiral galaxies. The 17th century mathematician Jacob Bernoulli was so fascinated by this beautiful shape, he called it the "spiral mirabilis" (miraculous spiral) and asked for it to be engraved on his tomb stone, along with the sentence "Although changed, I shall arise the same." Unfortunately the engravers got it wrong and he ended up with an Archimedean spiral on his grave instead.
For our final shape, or rather family of shapes, let’s start by considering the equation
(The || symbol stands for the absolute value, so we are only considering positive numbers for .)
To see what is going on here, let’s first recall the shape of the sine function, plotting on the horizontal axis against on the vertical one:
Applying the absolute value means that the parts of the wave that are below the horizontal axis (for which is negative) are flipped to lie above it:
You can see that as moves from to , the value of moves from 0 to a maximum of 1 (at ) and back down to 0.
Now let’s go back to our polar coordinates. As ranges from to , the distance to of our points
ranges from 0 (at ) up to 1 (at ) and back to (at .) This creates a little loop in the upper half plane. Then as moves from to the distance varies in exactly the same way. This creates a little loop in the lower half plane.
Now let’s up the stakes and consider the equation
The new factor of 2 means that the two loops that previously appeared for ranging from to now appear for only ranging only from to . This means that both loops occur in the upper half plane and also become a little squashed. As moves on from to , the value of ranges from to Since the sine function is periodic (it has the same value for as it does for ) a mirror image of our pair of loops now also appears in the lower half plane:
We have a flower with four petals!
What happens if we consider the equation
where is any positive whole number? You’ve guessed it! You get a flower with petals. It’s amazing what polar coordinates can do!
The shapes given by r = |sin(kθ)| for k=3, k=4 and k=5 (from left to right)
About the author
Marianne Freiberger is Editor of Plus.