How to check if a box fits into another box (any rotations allowed)
algorithm, computational-geometry, geometry, language-agnostic, math
Solution
Not a complete answer, but a good start is to determine the maximum diameter that fits inside the larger box (inscribe the box in a circle) and the minimum diameter needed for the smaller box. That gives a first filter for possibility. This also tells you how to orient the smaller box within the larger one.
Problem
Suppose I have two boxes (each of them is a rectangular cuboid, aka rectangular parallelepiped). I need to write a function that decides if the box with dimensions (a, b, c) can fit into the box with dimensions (A, B, C), assuming any rotations by any angles are allowed (not only by 90°). The tricky part is the edges of the inner box may be not parallel to corresponding edges of the outer one. For example, a box that is very thin in the dimensions (a, b) but with length 1 < c < √3 can fit into a unit cube (1, 1, 1) if placed along its main diagonal. I've seen questions [1], [2] but they seem to cover only rotations by 90°.