However, this kind of motion not only appears in the motion of wheel, but also in one of my favorite sports -10m platform.
Check women's 10m platform FINA/NVC World Diving series London 2014 online
Of course, the jumping of the athletes is very gorgeous, fluent, and with very complicated rotation and gestures. I begin to wonder, why the motion of the athlete is a very perfect parabola even though she made so many rotations and changes in gesture during the jumping?
In fact, this motion is very similar to the motion of the wheel.
The athlete keeps rotating during the whole process. For the transitional motion, the only force causes her to move is gravitational force, which causes the parabola-like motion. But for the rotational motion, no matter how the athlete rotates, the rotation only changes the velocity of her body parts except the centre of mass. According to the motion of the wheel, the centre has 0 angular velocity, so does the centre of mass of the athlete here. That's why when the athlete jumps, she follows a perfect parabola, because her rotation does not affect the motion of the centre of mass.