How to calculate all 24 rotations of 3d array?

Question

I have a 3d numpy array describing a polycube (imagine a 3d tetris piece). How can I calculate all 24 rotations?

Numpy\'s array manipulation routines includes a rot90 me

Answer 1

There are 24 rotation matrices at the bottom of this page: http://www.euclideanspace.com/maths/algebra/matrix/transforms/examples/index.htm .

If a tetris piece were represented by an array of coordinate triples, to apply a rotation to the piece you would just matrix multiply the matrix by each of the triples in the array. You would do that 24 times to get the 24 different pieces.

For example, if your array of boxes is (1,2,5), (1,2,4), (1,3,4)

And the rotation you are considering at the moment is for example

    1 0  0
M = 0 0 -1
    0 1  0

Then the rotated figure has representation

(1,2,5).M, (1,2,4).M, (1,3,4).M

where the . represents matrix multiplication. Do that for each M in the list and you've got all 24 rotations. (You can either pre- or post- multiply by the matrices, it doesn't matter if you want the whole set of rotations in no particular order.)

So that is very straightforward.

Now, to get the data out of your data structure and into the array structure I have used above, you would have to do something like (sorry I don't know Python)

for i = 0 to 2
  for j = 0 to 2 
    for k = 0 to 2
      if polycube(i,j,k)==1 then push(i-1,j-1,k-1) onto end of newArray

or something like that. Finally, to go from the newArray representation to the polycube you would do something like

  polycube = all zeros
    foreach point (i,j,k) in newArray
      polycube(i+1,j+1,k+1) = 1

For a 2x2x2 polycube, you would do

for i = 0 to 1
  for j = 0 to 1 
    for k = 0 to 1
      if polycube(i,j,k)==1 then push(i-0.5,j-0.5,k-0.5) onto end of newArray

and the corresponding inverse function.