Imagine [color=#0000ff]Earth (blue point)[/color] and [color=#ff00ff]Venus (pink/magenta point)[/color] revolving in a common circular orbit with the Sun at the center (origin/pole). Imagine that for every 2 orbits of Earth, Venus completes 1 orbit in the same time period ("orbit_ratio" slider). Press the "start/stop m" button to see this in action. Click on the "trace" checkbox to see the interesting geometry that forms from the segment drawn between the two planets throughout their orbits.[br][br]The "[color=#0000ff]Earth[/color]/[color=#ff00ff]Venus[/color]" button alters the settings to reflect actual ratios of orbital radii and periods for Earth and Venus. However, this model assumes concentric circular orbits in a common plane rather than the actual slightly-elliptical planetary orbits. Again, animate with "trace" on and see what interesting geometry results. Do you see how the 13:8 orbital period is reflected in the geometry?[br][br]Play with the other settings to observe how they affect the geometry. To precisely increment the sliders on a computer, it may be easiest to click on the slider and then use the left/right arrow keys on your keyboard.[br][br]~ ~ ~[br][br]I created the first version of this construction in 2018 when a student sent me a similar Earth-Venus graphic that he saw on Instagram. In 2024, I came across this [url=https://twitter.com/MAstronomers/status/1855467924066336846]post[/url] that does something similar with Saturn, so I then added the [color=#0000ff]Earth[/color]/[color=#ff00ff]Saturn[/color] button and updated the construction to allow for a greater variety of settings to accommodate it.[br]

When R[sub]ratio[/sub] = 1 and the "curve" box is checked, the displayed epicycloid will perfectly match the geometry bounded by the AB line segments.[br]When R[sub]ratio[/sub] ≠ 1 (points A and B orbit circles of different radii), then the bounded shape is not truly an epicycloid. A checked "curve" box will display (in a muted color) the epicycloid that most closely resembles the bounded shape.[br]