This animation is made by Idan Tal and I found it on his twitter page. I hid it behind a spoiler tag because the animated gif might annoy some people.
Let's say the cardioid is described by the polar equation $r=1-\sin\theta$, and the center of the red disk is located on $(0,-\frac12)$ whose radius is $\frac12$. The circle is rotating around the red disk with radius $1$. Now it looks like the moving circle always intersects the cardioid in at least two points that are colinear with its center. In other words, the cardioid is the geometric location of the endpoints of a specific diameter of the moving circle (or so it seems).
I am struggling to find a simplified formula for this diameter. I tried writing the parametric equations for the moving circle and then solving its intersection points with the cardioid, but the equations (in Mathematica) became highly complex of degree $8$ or more. And there is no chance in heaven that the equations could be solved. I am wondering, is there any simpler approach for this? How can we find the coordinates of those two points, or maybe the equation of their connecting line?