How to calculate the angle of rotation of a gear when hitching with another gear?

Question

Geometry

I am developing a logic game about gears, I ran into the problem of accurately coupling the gears so that the teeth fall correctly between each other.
2D space, two gears, diameters are different (number of teeth too). To the first we cling to the second. The first will have parameters in capital letters. The second one is small. R and r - radii, number of teeth T and t, W and w - angle of rotation (360 degrees - full circle).
The whole snag in order to take into account the relative position of the X, Y centers of the gears depends on how much you need to additionally tighten the angle so that the teeth do not fit each other.
Here are pictures to illustrate what is meant:

Answer 1

Shouldn't the linear speed of the teeth (in fact, circles) at the point of contact of the gears be equal to ? Otherwise, one of the shafts will be required to constantly rotate with slippage ...

Answer 2

I added pictures to the question, maybe it will be clearer what I want. I will reformulate it in a different way: at the moment of coupling, with further movement of the gear with a finger, it is no longer possible to move the gear to match the movement of the finger - freely in x, y. the minimum distance R+r must be maintained. Further movement is possible along an arc with a rotation by a certain angle of the gear itself. - this is the dependence - the formula I need.