What's the difference between these rotations?

1) Each point on the coordinate plane is rotated $\theta$ degrees about the origin.

2) Each point $P$ with the coordinates $(x,y)$ is rotated $\frac{\pi}{4}$ radians about the origin.

The answer says rotation 2 "defines some strange transformation that doesn't preserve angle measures or segment lengths."

I don't see how this rotation is different than the first one and how the second rotation causes the weird transformation. Could someone provide a more detailed explanation?

• Well, what's the size of $\theta$?
– Newb
Commented Jul 13, 2014 at 0:37

The first transformation clearly is just a rotation around the origin of the entire plane with angle $\theta$.
The second transformation could be a rotation around the origin of the entire plane with angle $\frac \pi4$. But when $x$ en $y$ are given, the second transformation only moves (rotates) only one point of the plane, and thus lengths are not preserved.
• That's another way I've considered it, but how do you end up with a rounded curve when only one point moves? Wouldn't you end up with the original line except with that one point rotated at $\frac{\pi}{4}$? Commented Jul 13, 2014 at 14:35