1- In the literature, I found transformations apply all points in the plane. Consider a rotation of a triangle. When we rotate the triangle we apply the rotation not just for the triangle, but for all points in the plane. In this case, are we rotating the plane as well as the triangle?

2- Or is the plane is fixed and we are just identifying the corresponding points for the pre-image points? If yes, Can we call this the mapping of the points of the plane? In this case can we talk about the motion at all?

I would appreciate if you share your thoughts with me.

M&M

Yes, when we talk about rigid transformations, the intention is to a trasnformation of the plane. The whole plane moves.

>>infinite number of discrete points]

I am not sure about that.

>as well as the triangle?
>>and we specified the coordinate axis and know where the
>origin is]

You may choose coordinate axes at will, yes. But you may also choose to move the coordinate axes along with the plane.

>
>2- Or is the plane is fixed and we are just identifying the
>corresponding points for the pre-image points?

Usually, the idea of rigid motion applies to the plane as a whole.

>If yes, Can
>we call this the mapping of the points of the plane?

If the intention is to all points of the plane, but the transformation is only defined for points of a triangle, then, no, I do not think it is customary to call such a transformation "transformation of points of the plane." However, I can imagine that somebody does use this term in a non-inclusive manner, i.e., not meaning "transformation of all the points in the plane."

> In this
>case can we talk about the motion at all?

We are, are we not? (I mean, you and I, and right now.)

To boot, if a train moves on the plane, it may be a convenience to consider a motion which could be applied to a whole plane, but, for a task at hand, applies only to its subset.