How would I translate screen coordinates to Isometric coordinates?

Question

I have a function isoToScreen(x, y) that converts Isometric coordinates to Screen coordinates.

var tileW = 16;
var tileH = 16;

var isoToScreen = function(x, y) {
    var posX = (x - y) * tileW;
    var posY = (x + y) * tileH / 2;

    return [posX, posY];
};

But how would I make a function that converts screen coordinates back to Isometric coordinates?

var pos = screenToIso(16, 8); 
pos[0] = 1; // Iso X
pos[1] = 0; // Iso Y

Answer 1

var screenToIso = function(screenX, screenY)
{
  var isoX = screenY / tileH + screenX / (2*tileW)
  var isoY = screenY / tileH - screenX / (2*tileW)

  return [isoX, isoY];
}

To get this function you need to rewrite the original math

screenX = (isoX - isoY) * tileW
screenY = (isoX + isoY) * tileH / 2

Starting with the first line you get the following:

screenX = (isoX - isoY) * tileW
screenX / tileW = isoX - isoY
screenX / tileW + isoY = isoX

The second line:

screenY = (isoX + isoY) * tileH / 2
2*screenY = (isoX + isoY) * tileH
2*screenY / tileH = isoX + isoY
2*screenY / tileH - isoX = isoY

Now the two lines look as follows:

A) isoX = screenX / tileW + isoY
B) isoY = 2*screenY / tileH - isoX

Then substitute isoY in the line A, with the formula derived from line B:

isoX = screenX / tileW + 2*screenY/tileH - isoX
isoX+isoX = screenX / tileW + 2*screenY/tileH
2*isoX = screenX / tileW + 2*screenY/tileH
isoX = screenX / (2*tileW) + screenY/tileH
isoX = screenY/tileH + screenX / (2*tileW)

And finally, substitute isoX in line B, with the formula derived from line A:

isoY = 2*screenY / tileH - screenX / tileW - isoY
2*isoY = 2*screenY / tileH - screenX / tileW
isoY = screenY / tileH - screenX / (2*tileW)

Solving linear equations comes very much in handy for a lot of programming, especially graphics or game related programming. Pick up a book on Algebra or read some online tutorials, you will thank yourself later.

Answer 2

Just reverse the order of operations:

var tileW = 16;
var tileH = 16;

function screenToIso(x, y)
{
    var posX = ((y * 2 / tileH) + (x / tileW))/2;
    var posY = (y * 2 / tileH) - posX;

    return [posX, posY];
}