Detect if one rect can be put into another rect

Question

This problem is different from testing if one rect is in another rect.

Known information is the sides length of two rects.

How to calculate if one rect can be pu

Answer 1

The first check one would do is of course whether the rectangle fits inside the other in either of the axis aligned orientations.

If not, the only option for it to fit is diagonally, but there might actually be many angles for which it fits, the difficulty is, not just guessing but indeed calculating a possible angle, if one exists.

Now, notice that if the inner rectangle does indeed fit diagonally, then you can rotate it until two if its opposite corners touch either both the top and bottom edge of the outer rectangle, or the left and right. (In your diagram more or less the first.)

In that case you already know that you have fit it inside in that one dimension(in the example, the y-axis). You then have to calculate the bounding width of the inner rectangle in the other dimension and check that against the width of the outer box.

There might be a smarter algorithm out there for this, but I am 100% sure that what I describe works. Let me know if you can figure out the math for this yourself(if you think this is a good solution), if not, I might have a go at it later. I wonder if my algorithm can be implemented completely without trig functions...

EDIT: Ok now, I could not resist...

Here is the math I did to solve the problem as outlined above: (Sorry, only in image form, I hope my handwriting is readable.) I would be happy if someone could check my math. I do not see anything wrong with any of the steps right now, but it is always better to have someone else check. (And of course: Use this at your own risk.)

If anyone finds anything wrong with this algorithm, please let me know and I will fix it as soon as possible.

Also I would be highly interested to see if anyone has a better solution, involving less complicated math. Maybe a vector based approach?