Rolling Wheel Creates Cycloid and Brachistochrone
This animation solves the differential equation for the curve (cycloid) of
least time between two points and also shows the rolling wheel that generates that curve.
The physicsanimation.org menu item just below this item allows the learner to discover that
the cycloid curve is indeed the one of least time.
The differential equation for the motion of a frictionless bead on the cycloid is very simple:
dy/dx=sqrt(h/y-1)
where h is the maximum value of y (the total fall height).
The equations for the motion of a point at the outer radius of the wheel are:
xPoint=r*(phi=cos(phi))
yPoint=r*(1-cos(phi))
where r=h/2 and phi is the angle of rotation of the wheel.
The green curve and disc represent the solution to the differential equation.
The red curve and the red circle at the edge of the wheel are the curve traced by the center
of the red circle as the wheel rotates on a horizontal line above it.
I have offset the y and x positions of the green curve by 5 pixels to avoid overwriting curves.