vector - Adobe illustrator perfect curve between two grids

How would you combine a vertical grid and horizontal grid with matching width & heights with perfect curved lines?

Example:

I'm trying to find a method that would be salable for 100, 1000, etc grids.

Answer

I would solve that by math:

The "perfect" curve means (for me) square based curve, so the task is to find the squares, create related to them arcs and make a blend in N of steps as need.

I've created 2 guides and 1 square between one square and guides intersection -

Now I created an arc as shown and with pen tool elongated it to the second square.

The same operations for a larger arc -

And blended the arcs created -