Algorithm for iterating over an outward spiral on a discrete 2D grid from the origin

Question

For example, here is the shape of intended spiral (and each step of the iteration)

          y
          |
          |
   16 15 14 13 12
   17  4  3  2 11
--         


        

Answer 1

This problem is best understood by analyzing how changes coordinates of spiral corners. Consider this table of first 8 spiral corners (excluding origin):

 x,y   |  dx,dy  | k-th corner | N | Sign |
___________________________________________
1,0    |  1,0    | 1           | 1 |  +
1,1    |  0,1    | 2           | 1 |  +
-1,1   |  -2,0   | 3           | 2 |  -
-1,-1  |  0,-2   | 4           | 2 |  -
2,-1   |  3,0    | 5           | 3 |  +
2,2    |  0,3    | 6           | 3 |  +
-2,2   |  -4,0   | 7           | 4 |  -
-2,-2  |  0,-4   | 8           | 4 |  -

By looking at this table we can calculate X,Y of k-th corner given X,Y of (k-1) corner:

N = INT((1+k)/2)
Sign = | +1 when N is Odd
       | -1 when N is Even
[dx,dy] = | [N*Sign,0]  when k is Odd
          | [0,N*Sign]  when k is Even
[X(k),Y(k)] = [X(k-1)+dx,Y(k-1)+dy]

Now when you know coordinates of k and k+1 spiral corner you can get all data points in between k and k+1 by simply adding 1 or -1 to x or y of last point. Thats it.

good luck.