Rolling_On_Parabola

Rolling of a wheel w inside a parabola p

p the parabola, defined by its directrix d and its focus F.
p' is the parallel-curve to the parabola, generated by the centre A of the wheel, as it rolls on the parabola. The parallel-to-conics curves are never conics and they can have cusps. To see this, switch to the [selection-tool] (press CTRL+1) , catch point F and move it towards d. After a while the curve p' (which is always parallel-to-conic p) gets cusps near F. The weel can freely roll on the parabola, if and only if p' has no cusps.

Press the green button to start the rolling motion of the wheel. The press the red button to stop it.
Switch to the [pick-move-tool] and catch point A (middle of wheel) move it slowly to increase/decrease the radius of the wheel. Watch the movement of the wheel, when it comes near to F. While the weel moves, switch again to the selection-tool, catch F and move it towards d, until the cusps of p' appear.

You can see the various parallel-to-conic curves p', by switching to the [pick-move-tool] (press CTRL+2) and pick-moving point B. This changes the vertical distance k = BB' of the parallel p' from p.

For a related subject look at the example look at the document: Parallels_to_Ellipse.html .

http://users.math.uoc.gr/~pamfilos/