So the kinematics of the contact point of a disc rolling without slip in a Cartesian plane if fairly straightforward. The velocity for the contact point is just v = ω r ev where r is the wheel radius and ev the current direction vector of the wheel.
However, say that you grab the plate and force a rotation around another axis than around where it would roll when rolling freely on the floor. If the plate is vertical, the contact point will not change (if anything due to you drill a hole in the table after while perhaps). However, if you lean the plate, this rotation will clearly induce a change of the contact point. The rate of change is largest when the plate is almost flat to the surface. Does that make sense? I could not find any visualization for this, nor did I bother to formulate this formally as I do not think it helps too much, but if you grab a plate you could probably figure out what I mean.
Where can I find a model for this kinematics? I am modeling a unicycle and no papers I have read so far consider this effect. Probably the answer is related to the projection of the disc onto the plane.
hint:
A wheel, tilted wrt the normal to the plane, will move same as a circular cone having the wheel as (a) base, and axis the wheel axis, so that the cone vertex is at the point where the axis meets the plane ...