/ic/ - Artwork/Critique

I'm trying to learn perspective but almost no resource I find gives a formal, mathematical approach, which I seek to develop intuition properly.

In this case, I want to rotate this cube by its vertical axis from the angle in red at 45 degrees to the angle in blue, I know where the new diagonal should lie, but I don't know how to keep the scale of the cube consistent, I don't know where to cut the line to get the cube to be the same size but in perspective. Does anyone here know how?

I really don't understand either the reasoning for measuring the distance between a vanishing point and a station point and then drawing it over the horizon. I see it works but I don't know why.