Wheels within wheels: A gallery of epicycles

Modelling the solar system

If you look at the night sky, the majority of the stars stay in the same position relative to each other, but others are more interesting -- they tend to wander around, sometimes moving one way, and sometimes the other:

Sorry, the GeoGebra Applet could not be started.

In this animation the blue circle is moving at constant angular velocity, while the white circle represents the more complicated apparent motion of a planet.

This variable motion (technically called prograde and retrograde motion) caused problems to people trying to create early models of the solar system. Obviously (to them) the Earth needs to go at the centre, and obviously the planets should travel in circles around the Earth (as the circle is the most perfect shape), but this doesn't account for the motion we see.

Around 2000 years ago, the astronomer Ptolemy introduced the concept of epicycles to explain the motion of the planets. Select all the 'Show' check boxes in the animated environment above, and you will see the following:

The large blue circle is the deferent, and the pink one, which is centred on the deferent, is the epicycle. In our model we have a third, smaller circle, centred on the epicycle, and futher complicating the motion of the planet. This was not a feature of Ptolemy's original model of the solar system (which used 34 circles), but were sometimes introduced in more complicated refinements of the model.

Each of these circles move at different rates, and it's the movement of these circles that, in Ptolemy's model, causes the variations in motion (and in brightness) of the planets. You can control the size and speed of each of these circles using the sliders at the bottom of the environment. You can also pause/restart the animation, choose which of the circles should be drawn, and draw the orbit of the planet (there will be more on orbits in the next section!).

Note that both of the epicycle circles are much larger than those that Ptolemy used -- by far the most important component of motion was given by the deferent.

Orbits and apparent motion

After we have set up a system like this, one natural question to ask is: so, what does the orbit of our planet look like? Selecting the 'Orbit' option, we see the following amazing path appear:

My focus for the rest of this post is going to be on these orbits, rather than using epicycles to model the solar system. If you're interested in the actual Ptolemeic model, here are two excellent sites for further exploration.

Before we move on to just looking at the orbits, though, I should point out that it can be very difficult to figure out what the size and speed of the epicycles should be, given the apparent motion (which is what you actually observe). Going the other way can be interesting, as well. Here are two examples of settings to explore:

First, (1,1), (0.5, -1), (0.5, 3). Graphically,

This has orbit:

Second, (1, 1), (1, -1), (0, 0), with orbit

What will the apparent motion of the planet be for each of these? Set them up in the interactive environment at the start and see if you were correct!

A gallery of orbits

Here's a simplified version of the environment above, concentrating on the epicycle orbits:

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

The parameters are exactly the same as for the original animation: the size and speed of the three generating circles.

Playing around with this a while, you find that altering the 'size' parameters tends to continuously alter the final orbit, while tiny changes in the 'speed' parameters can lead to major changes in the orbit. It's actually even worse than it seems! I have deliberately chosen the 'speed' divisions to guarantee that the orbits will be closed, after a maximum of 10 revolutions. Can you see how? If so, you should be able to figure out which combinations of settings give you orbits that repeat after 2, or 5 revolutions of the first circle.

If we happened to choose an irrational value for one of the speeds, then the orbit would never exactly repeat, and we would end up with our orbit picture looking like a filled doughnut.

Here are some of my favourite orbits:

(Click on these to see them at a larger size.)

All of them were generated by keeping the size/speed of the first circle as 1 and 1, and only varying the other four parameters. It's amazing the variation in shapes you can get from such a simple set up.

If you are interested in printing these out, I have made a PDF of the orbits: epicycle-orbits.pdf. You can also download SVG files of these orbits, for editing in a vector drawing package such as Inkscape: epicycle-orbits.zip.

I have deliberately not told you the parameters for these, although the file names will give a hint! I would love to see any other interesting patterns that people find.

Orbits and Spirograph patterns

These epicycle orbits are related to (but not the same as) Spirograph patterns, more mathematically called hypocycloids and epicycloids:

These are generated by having circles rolling around the outside (or inside) of other circles. You can see the relationship if you compare the parametric equations for our curves:

$x(t) = r_a \cos( \theta_a t) + r_b \cos( \theta_b t) + r_c \cos( \theta_c t)$ ,
$y(t) = r_a \sin( \theta_a t) + r_b \sin( \theta_b t) + r_c \sin( \theta_c t)$ .

with those for 2-wheel Spirographs, with circles of radius $R$ and $r$ :

$x(t) = (R-r) \cos(t) + p \cos ( \frac{R-r}{r} t )$ ,
$y(t) = (R-r) \sin(t) + p \sin( \frac{R-r}{r} t )$ .

If you are interested in plotting Spirographs, a great link is this very detailed Linux Gazette article explaining how to plot them using Gnuplot.

The interactive environments above are written using Geogebra. Please feel free to download and play with either the full environment, or the orbit-only environment. Opening these files in Geogebra will let you explore exactly how I created them.

One Response to “Wheels within wheels: A gallery of epicycles”

Recent starred posts in my reader « The Number Warrior Says:
April 6th, 2009 at 2:26 am
[...] The Final Frontier, Wheels within wheels: A gallery of epicycles. Just [...]

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