Rotating rectangular array by 45 degrees

Question

Suppose:

2D array:  abcdef
           ghijkl
           mnopqr

Stored in simple string of length width * height, thus, let's call it arr.

arr = abcdefghijklmnopqr
width = 6
height = strlen ( arr ) / width

The goal is to rotate this array by 45 degrees ( PI/4 ) and get the following result:

arr = abgchmdinejofkplqr
width = 3
height = 8
converted to 2D array:  a..
                        bg.
                        chm
                        din
                        ejo
                        fkp
                        .lq
                        ..r

I have spent a few hours trying to figure out how to make this conversion and came up with a few semi-functional solutions, but I can't get it fully working. Can you describe / write an algorithm that would solve this? Preferably in C.

Thanks for any help

Edit: This is what I've tried already
Edit2: The purpose of 45 degree rotation is to turn diagonals into lines so that they can be searched using strstr.

// this is 90 degree rotation. pretty simple
for ( i = 0; i < width * height; i++ ) {
  if ( i != 0 && !(i % height) ) row++;
  fieldVertical[i] = field[( ( i % height ) * width ) + row];
}   

// but I just can't get my head over rotating it 45 degrees. 
// this is what I've tried. It works untile 'border' is near the first edge.

row = 0;
int border = 1, rowMax = 0, col = 0; // Note that the array may be longer
// than wider and vice versa. In that case rowMax should be called colMax.

for ( i = 0; i < width * height; ) { 
  for ( j = 0; j < border; j++, i++ ) {
    fieldCClockwise[row * width + col] = field[i];
    col--;
    row++;
  }

  col = border;
  row = 0;
  border++;
}

The 'border' in my code is an imaginary borderline. In the source, it is a diagonal line that separates diagonals. In the result, it would be a horizontal line between each row.

1   2   3 / 4   5
6   7 / 8   9   10
11 /12  13  14  15

Those slashes are our borderline. The algorithm shoul be pretty simple and just read riagonals: first number 1, then 2, then 6, then 3, then 7, then 11, then 4 and so on.

Answer 1

I would like take help of above and solve this by an example:

import pandas as pd
import numpy as np
bd = np.matrix([[44., -1., 40., 42., 40., 39., 37., 36., -1.],
                [42., -1., 43., 42., 39., 39., 41., 40., 36.],
                [37., 37., 37., 35., 38., 37., 37., 33., 34.],
                [35., 38., -1., 35., 37., 36., 36., 35., -1.],
                [36., 35., 36., 35., 34., 33., 32., 29., 28.],
                [38., 37., 35., -1., 30., -1., 29., 30., 32.]])
def rotate45(array):
    rot = []
    for i in range(len(array)):
        rot.append([0] * (len(array)+len(array[0])-1))
        for j in range(len(array[i])):
            rot[i][int(i + j)] = array[i][j]
    return rot

df_bd = pd.DataFrame(data=np.matrix(rotate45(bd.transpose().tolist())))
df_bd = df_bd.transpose()
print df_bd

of which output will be like:

44   0   0   0   0   0   0   0   0
42  -1   0   0   0   0   0   0   0
37  -1  40   0   0   0   0   0   0
35  37  43  42   0   0   0   0   0
36  38  37  42  40   0   0   0   0
38  35  -1  35  39  39   0   0   0
0   37  36  35  38  39  37   0   0
0    0  35  35  37  37  41  36   0
0    0   0  -1  34  36  37  40  -1
0    0   0   0  30  33  36  33  36
0    0   0   0   0  -1  32  35  34
0    0   0   0   0   0  29  29  -1
0    0   0   0   0   0   0  30  28
0    0   0   0   0   0   0   0  32

Answer 2

I'd call this diagonal scanning, not rotation by 45 degrees.

In your example, the diagonals are running down-left; we can enumerate them 1, 2, ...:

123456
234567
345678

This will be the counter for the iterations of the outer loop. The inner loop will run 1, 2 or 3 iterations. In order to jump from one numbered symbol to the other, do col--; row++; like you did, or add width-1 to a linear index:

....5. (example)
...5..
..5...

Code (untested):

char *field;
int width = 6;
int height = 3;
char *field45 = malloc(width * height);
int diag_x = 0, diag_y = 0; // coordinate at which the diagonal starts
int x, y; // coordinate of the symbol to output
while (diag_y < height)
{
    x = diag_x; y = diag_y;
    while (x >= 0 && y < height) // repeat until out of field
    {
        *field45++ = field[y * width + x]; // output current symbol
        --x; ++y; // go to next symbol on the diagonal
    }
    // Now go to next diagonal - either right or down, whatever is possible
    if (diag_x == width - 1)
        ++diag_y;
    else
        ++diag_x;
}

If you want to "rotate" in another direction, you may want to change ++ to -- around the code, and maybe change bound checking in the loops to opposite.

In addition, you can replace (x,y) coordinates by just one index (replacing ++y by index+=width); i used (x,y) for clarity.

Answer 3

I looked at http://en.wikipedia.org/wiki/Shear_mapping for inspiration and produced this python code:

a = [['a', 'b', 'c', 'd', 'e', 'f'],
     ['g', 'h', 'i', 'j', 'k', 'l'],
     ['m', 'n', 'o', 'p', 'q', 'r']]

m = 1 # 1/m = slope

def shear_45_ccw(array):
    ret = []
    for i in range(len(array)):
        ret.append([0] * 8)
        for j in range(len(array[i])):
            ret[i][int(i + m * j)] = array[i][j]
    return ret

print(shear_45_ccw(a))

produces:

[['a', 'b', 'c', 'd', 'e', 'f', 0, 0], 
 [0, 'g', 'h', 'i', 'j', 'k', 'l', 0], 
 [0, 0, 'm', 'n', 'o', 'p', 'q', 'r']]

which is something like what you want. The algorithm is hopefully readable even though its in python. The meat of it is this: ret[i][int(i + m * j)] = array[i][j]. Good luck! I cheated when I initialized the array; you'll have to deal with that differently in C anyways.

EDIT: also, I had no idea why your result was flipped and stuff: I trust you can make the right behavior happen.