Composite of two rotations with different centers

Question

What is the composite of two rotations having different centers? What type of transformation (translation, rotation, etc) is it? Or there is no rule for it?

Answer 1

Rotations are plane isometries and there are four types of plane isometries:

• rotations;
• translations;
• reflections;
• glide reflections.

The first two types preserve orientation, whereas the last two reverse it. Since the composition of plane isometries is again a plane isometry, and since rotations preserve orientation, the composition of two rotation can only be either a rotation or a translation. It turns out that it is a translation if and only if the sum of the rotation angles is a multiple of $360^\circ$.

Answer 2

Try out a few examples and see what happens. What about $90^\circ$ clockwise about the origin, and then $90^\circ$ counterclockwise about $(0,1)$? What if the last rotation is clockwise? What if it's $45^\circ$? What if the first rotation is $45^\circ$? Does it look nicer if the second point isn't $(0,1)$ but instead the point that the first rotation moves $(0,1)$ to? What if you pick a different point for the second rotation? Some experimentation like this ought to lead you to concrete ideas that might possibly be provable.

This kind of experimentation is how new math is made, and while I'm certain that the answer to your problem is known, you shouldn't let an opportunity like this to make your own, genuine discovery go to waste.