I want to understand what changing coordinate system means for hands of clock. Let's say the clock only has hour and minute hand. It can move let's say just in the upper 180 deg. of the clock (as shown in the figure). The area between the two hands is V1, and the rest is V2. Depending on the angles a and b, the area of V1 can be from 0 to Pi*r. This means, both independent hands of clock will contribute to the area, as can be seen in picture from 1) to 3). But I want to understand what kind of joint rotational modes is achieved by changing the coordinate system to a-b and a+b. How do I understand this physically? I understand a-b will give the angle in between the two clock. But what about a+b? a-b is sort of a stretch mode I guess? And is a+b the joint rotation mode (meaning both the hands move around at the same time, by the same angle). In addition, the reference will change too right, in these two coordinate system? How do I understand this change of coordinate system in the context of dividing the circle into two arbitrary regions V1 and V2? Thanks a lot!