The cycloid is defined by these equations:
c(t) = [t + L*cos(t), L*sin(t)]
The point where the wheel touches the street is momentarily at rest.
The connection to the curve point is a generalized radius and the curve
tangent is orthogonal to this radius. This is the same for all rolling
constructions. See for example
Epi- and Hypocycloids
The demo shows at each moment two consecutive positions of the dots. Therefore
the image does not look random but
suggests a "rotation pattern" of the dot velocities.
The random dots attached to the rolling wheel are intended to illustrate
how the plane of the wheel is, at each moment,
performing a rotating movement around the point where the wheel touches
the street.
We can choose any point of this plane as drawing pen and the velocity
vector of the rotating motion is the tangent
vector of the curve which the pen draws.