Calculate the planet's periode

Last night I was fascinated to find some planets on the sky and I wondered how to calculate their period in order to be able to find them again in the sky some months from now. This is a question already answered, but I just wanted to answer it myself in a very clear and simple way. It is not difficult at all, you just need to use common sense.

The planets and the Earth move in a way that can be compared to that of a wall clock. They rotate around a common center (the sun). Outer orbits possess longer periods of rotation around the sun than inner orbits. Having said that, now we can assume the outer planet behaves like the hour and the inner one behaves like the minutes. When the inner planet completes one revolution, the outer planet has just advanced a fraction of revolution(when the minutes complete one revolution, the hours reach an hour). After that, the inner planet advances and the line between the inner planet intersects that of the outer planet at time T(minutes and hours meet). How to calculate T ?

Let I be the interior planet's period of revolution around the sun.
Let O be the outer planet's period of revolution around the sun.

Inner Angle = 1Revolution * T / I
Outer Angle = 1Revolution * T / O

We need to find T such that :

Inner Angle - 1 Revolution = Outer Angle

1Revolution * T / I - 1 Revolution = 1Revolution * T / O

We can cancel out the 1 Revolution.

T/I -1 = T/O
T/I - T/O = 1

hence, we have T = 1 / (1/I - 1/O)

We just made an assumption: I < O. It does not have to be like that, in fact the wall clock is exactly the opposite case (minutes go faster that hours). But it doesn't really matter, as you can say I goes for minutes and O for hours, for example, and the calculations remain valid. To avoid any conflict, we may rewrite T as

T = 1/ (1/min(P1, P2) - 1/max(P1, P2)).

where min (P1,P2) is the minimum among P1 and P2, max(P1, P2) is the maximum among P1 and P2 , P1 and P2 are the Periods for planets 1 and 2 respectively.

This calculations were performed initially by Copernicus. I hope the wall clock simile helps clearly understand what happens there.