Brian sent us his thoughts on this
What happens: It is impossible to tell from the motion of the plank
that the roller is not circular. The plank moves forwards at a
steady rate (assuming a steady push!) and remains at a constant
height (the side length of the triangle) above the floor.
Why: Consider a starting point with C on the floor and the plank
parallel to the floor and touching the roller at B (BC is
vertical). Move the plank, rolling the roller, until the plank is
touching the roller at A (so now AC is vertical). The roller will
be rotating about C. Clearly, given that curve AB is equidistant
from C, the plank will remain at a constant height above the floor.
The same arguments can be used if a vertex is against the plank and
the circular section is against the floor; only the frame of
reference is changed and relative motion of plank and floor are the
same.At the point of transition, both cases
are simultaneously true.
Now consider the transition. In this example, C will be on the
floor, A will be against the plank, and any further movement will
result in A being effectively fixed on the plank and curve BC
rolling on the floor. Note that the tangent to curve BC at C is
parallel to the tangent to curve AB at A --- AC is a common radius
of both curves. Hence at the transition there will be smooth
transfer of movement from one mode to the other. Despite one's
intuitive notions, the roller will therefore have a smooth forward