Earth-Venus cardioid-like orbit patterns
Imagine Earth (blue point) and Venus (pink/magenta point) revolving in a common circular orbit with the Sun at the center (origin/pole). Imagine that for every 1 orbit of Venus, the Earth completes 2 orbits in the same time period ("coeff" slider). Press the "animate" 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. Now press the "Venus" button, which alters the settings to reflect more accurate ratios of orbital radii and periods for Earth and Venus. Again, press animate (with "trace" on) and see what interesting geometry results. Do you see how the 13:8 orbital period is reflected in the geometry? 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.