Isometric Drawing Tool: Converting 2D information to 3D

Question

I was drawing with the NCTM Isometric Drawing Tool, and produced the image seen here. I also noticed that it is possible to view the isometric drawing in 2D, and was wondering if/how it is possible to generate the 3D-coordinates of the blocks by using the information given in 2D.

Answer 1

Isometric projection, in this case, is the orthogonal projection on plane with equation $x+y+z=0$ (or any affine plane with equation $x+y+z=d$). Thus

A first answer is 'no' because depth information (depth in the direction of view, i.e., in the $(1,1,1)$ direction) is lost ; in the example you give, one could interpret the upper corner of the highest cube as having coordinates $(0,-1,5)$ but they could as well be

$$(0,-1,5)+t(1,1,1)=(t,t-1,t+5)$$

for any real number $t$...

A second answer is 'yes' as long as we have the absolute coordinates $(a,b,c)$ of a certain reference point $R$. Indeed in this case, one can retrieve the absolute coordinates of any point $S$ by first creating a path from (the projection) of $R$ to (the projection of) $S$ and then, while following the path, add $(\pm 1,0,0), (0,\pm 1,0), (0,0,\pm 1) $ to the current coordinates as we are moving South-East, North-West, South-West, North-East, South or North direction resp.

Remarks :

1) all paths from $R$ to $S$ give the same final coordinates to $S$.

2) thinking to reference (affine) plane with equation $x+y+z=1$, one could work with barycentric coordinates. I can even imagine that this "Isometric Drawing Tool" has been programmed using sort of coordinates. I will try to re-create it in this way in the coming times...