I saw this post on Facebook illustrating the so-called "Dance of Earth and Venus" and thought it was pretty cool. Venus completes (close to) 13 revolutions for every 8 earth revolutions; I guess we could say that Venus' speed relative to Earth is 13/8. As they revolve around the sun, the line segment betweens them envelopes a curve with pleasant five-fold symmetry.
I was curious about what other types of patterns are possible so I created this visualization that allows you to play with speed ratio. Fractions like p/q, are nice. the Earth goes around q times while the interior planet goes around p times. Other numbers, like 1.23 or π (entered as pi) or the Golden Ratio (entered as phi) are interesting as well, but it's hard to say how long the animation will run for.
Questions for consideration
Here are a few questions about the relationship between the images we see and the parameters we input.
- If we enter a fraction p/q for the relative speed, how many revolutions does the blue planet make before the planets are aligned in their initial position?
- What happens if we enter an irrational number for the relative speed?
- What happens if we enter a negative number for the relative speed?
- How is the symmetry we see related to the relative speed that we enter? Does your description work for positive and negative numbers?
- How does the inner radius affect the image you see?