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Hopefully this is a good question. I am working with a native class that represents a 2D image as a 1D array. If you want to change one pixel, for example, you need to now how to derive the index from the x,y coordinates. And I am trying to grok, in abstract terms, how to map a multidimensional object to its underlying unidimensional representation.

So, let's say we have a 1D array array1d like this:

array1d = [ a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y ]

In the context of our program, array1d represents a 2D grid:

a b c d e
f g h i j
k l m n o
p q r s t
u v w x y

And we want to perform operations on array1d such as:

  • Get the value at x,y coordinates (in this example, 1,2 would give l)
  • Get any sub-grid using x,y,width,height (1,2,2,2 would give [l, m, q, r])
  • Set the value at any x,y coordinate (etc.)

Is there a computer science-related or mathematical term for said operations, which map points/regions in 1D data to coordinates in 2D/3D/4D... objects? If so, what is it? Clearly all computers with displays solve this same issue, so what is this fundamental problem called and how is it conventionally solved?

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In Matlab, and thus likely math types (which spills into CS), to convert one matrix into another (be it a 1x12 into a 2x6 or a 2x6 into a 3x4) is known as "reshaping" mathworks.com/help/matlab/ref/reshape.html –  MichaelT Sep 28 '13 at 17:08

1 Answer 1

up vote 22 down vote accepted

2D / 1D - mapping is pretty simple. Given x and y, and 2D array sizes width and height, you can calculate the according index i in 1D space (zero-based) by

i = x + width*y;

and the reverse operation is

x = i % width;    // % is the "modulo operator", the remainder of i / width;
y = i / width;    // where "/" is an integer division

You can extend this easily to 3 or more dimensions. For example, for a 3D matrix with dimensions "width", "height" and "depth":

i = x + width*y + width*height*z;

and reverse:

x = i % width;
y = (i / width)%height;
z = i / (width*height);
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+1 for simple algebraic solution! –  recursion.ninja Sep 28 '13 at 18:07
@awashburn that is the traditional way to do it, it's even built into compilers for static 2D arrays –  ratchet freak Sep 28 '13 at 18:26
Thanks for the answer (and the reverses)! Is there a name or category for these general types of formulae (which map 1D -> 2D -> 3D etc.)? –  GladstoneKeep Sep 28 '13 at 23:28
@mtoast: I don't think so, its just basic integer math. –  Doc Brown Sep 29 '13 at 19:11
This example is wrong for 3D. The word depth in the calculation should be height. –  jiggunjer Jul 22 at 11:15

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