As I digest the paper, I hope to follow suit with a page of my own. Meanwhile, here is an old dish with a new dressing. T. Initially they coincide. A remains on the stationary circle. M is the point of the moving circle that traces the cardioid.
T is the point of contact of the two circles.
The fact that the motion is without slipping is more accurately expressed as the congruence of the corresponding arcs TA and TM of the two circles. Arc congruence creates a shifting symmetry especially noticeable when the three points are close to each other. Because of that symmetry, the bisector is tangent to both circles. The midpoints of segments AM lie directly on the tangent lines. Therefore, the image of the cardioid under the transformation is a smaller cardioid, and we get a third definition of the cardioid as the pedal curve of the circle.