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I have isometric assets that unfortunately can't change and need to work out the projection for those. Here is how a tile looks like:

floor tile

The tile image width/height ratio is 0.(6) (height/widht = 1.5). Since this is isn't the standard(2:1) isometric ratio, how can I work out the iso-screen/screen-iso coordinate conversion functions ?

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Minor quibble: you should use the term "orthographic" for this type of projection. "Isometric" is a very specific subset of orthographic projections, where unit vectors on each axis project to the same length (iso metric = equal measure). A true isometric tile will have a width:height ratio of sqrt(3) : 1. If that's not your ratio, then you're not using isometric projection, just a similar-looking orthographic projection.… – DMGregory Mar 25 '14 at 17:56
True, I tend to use orthographic projection when I mean cylindrical as opposed to conical – George Profenza Mar 25 '14 at 20:04

Having recently done a pixel to tile, tile to pixel conversion I understand this can get difficult quickly. You have a ratio of 0.6, usually the ration is 0.5, is this correct?

Also there are a number of ways to draw your tiles, diamond (chess board), staggered (Civ type game) etc.

Here is an answer to a similar question made by coobird (better than i ever could)

Here is a more in depth article that you should be able to use the work out the answer you need.

Good luck :)

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Thanks for the neat answer! That ratio is getting to me. The image w/h ratio is 0.66666667 (~ 0.(6) ), but I selected the bounding rectangle of the image non transparent content and that has a ratio of ~1.275. I'm getting close, but because of the numbers that don't divide nicely I get a few pixels off every few tiles. The articles look really explained. I'll take my time and go through them. Will let u know how this goes – George Profenza Jun 19 '12 at 14:00
Thats fine George, i'm using other peoples answers here. These are the articles which helped me. I'm at work now so dont have my engine code here but when i get time i will see if i can solve the issue but if you have a ratio of 0.666666666 could you not use (2/3) to get your isometric ratio? Ive just seen that you have a larger destination rectangle with a definable ratio and an offset, so you should be able to use the standard formulas with offset tweeks. If you have further issues post more code and i'll have a play. – LoveofSnow Jun 19 '12 at 15:01
Richard, I've gone through the resources, written a lot of code and so far so good! I have to squeeze your brains for another tip though: The designers can't render to a 2:1 because some elements will get occluded by walls at that angle, so that's why the custom angles/ratios. The problem with custom ratios is when I do the conversion and place bitmaps, (because of rounding I presume), the issue is bitmaps don't always align properly: every few tiles there's a 1px offset. Any hints on what ratio I should use to eliminate or make this issue less visible ? Thanks ! – George Profenza Jun 21 '12 at 15:48

The best explanation I have found of Orthographic/Cartesian to Isometric conversion is from this article.

BLOCK_SIZE = [64, 32]

def orth_to_iso(x, y):
    ''' Converts cartesian to isometric cordinates '''

        xx = (x - y) * (BLOCK_SIZE[0]/2)
        yx = (x + y) * (BLOCK_SIZE[1]/4)
    except Exception:
        xx = -1
        yx = -1

    return [xx, yx]

def iso_to_orth(x, y):
    ''' Converts isometric to cartesian coordinates '''

        xx = (x / (BLOCK_SIZE[0]/2) + y / (BLOCK_SIZE[1]/2)) /2
        yx = (y / (BLOCK_SIZE[1]/2) -(x / (BLOCK_SIZE[0]/2))) /2
    except Exception:
        xx = -1
        yx = -1

    return [xx, yx]
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Not being able to see the code or textures its difficult but if its not the tile itself being occluded then split the textures into two, ie if its a tile with a building on then draw the tile (at a 2:1 ratio) then 'layer' the building on top of that tile. This should work and you should be able to hold that 2,1 ratio.

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