I'm attempting to speed up box-to-box collision checks by first testing if two spheres large enough to encompass each (respectively) would collide (because I'm using separating axis theorem and it's slower than I'd like it to be).
This is fine for equilateral boxes (ie, cubes) since I can just take two opposing corners and the line between them is the exact diameter of a sphere big enough to encompass the whole thing properly; and also the point in it's middle is the origin of the sphere. I add an epsilon just to avoid any glitches with colliding with the corners of the cube, but theoretically it shouldn't be necessary.
But for non-equilateral boxes, such as frusta, this method does not work. I've pondered it over and nothing I can come up with is a quick enough calculation for my taste. The best I've come up with is to take two opposing edges (diagonally through the box) from the near face to the far face, get their individual lengths and sum them; then do the same thing for the left-to-right faces and top-to-bottom faces. with these three sums, take the largest and just use that.
Anyway, that method works but it's too much calculation, involving 6 square roots. I need something faster and preferably more accurate. I've googled and haven't found much on this subject at all. I'd like the solution to work for boxes of any shape (as long as faces are valid) and frustums of unknown orientation (meaning the large face could be any side, not a known one).
It's also important to know the origin of the sphere (which is the dead center of the frustum). Which if there is no faster calculation, this is just the sum of all 8 verts divided by 8.