# What are the extended homogeneous coordinates in the EdDSA specification?

According to the EdDSA specification from the IETF:

For point addition, the following method is recommended. A point (x,y) is represented in extended homogeneous coordinates (X, Y, Z, T), with x = X/Z, y = Y/Z, x * y = T/Z

I'm unfamilar with extended homogeneous coordinates, and I'm used to seeing points as simply (x, y). Can anyone explain where the T and Z variables come from in the specification document?

UPDATE:

I found this answer on Stack Overflow that has some great detail:

• Did you see the articles on the page 48? – kelalaka Apr 17 '20 at 15:52
• I did have a look through that section, and especially the "EdwardsPoint" class on page 50, but I couldn't see what the T and Z variables are. – simbro Apr 17 '20 at 15:59

$$X$$ and $$Y$$ roughly correspond to affine $$x$$ and $$y$$, $$Z$$ comes from regular extended (also called projective) coordinates as a scaling and $$T$$ is the actually “new” element that contains $$X\cdot Y$$. In particular, section 3 of the paper by Hisil et al. introduces the extended homogeneous coordinates and how to convert to and from extended homogeneous coordinates.
• @simbro When going from affine to extended homogeneous, $Z$ is always $1$ and $T$ is always $x\cdot y$, yes. The inverse operation from extended homogeneous $(X:Y:Z:T)$ to affine $(x,y)$ is $x=X\cdot Z$, $y=Y\cdot Z$ and ignore $T$ entirely. – xorhash May 15 '20 at 15:32
• @simbro Correction, but I can't edit the comment anymore: $x=X\cdot Z^{-1}$ and $y=Y\cdot Z^{-1}$, note that we take the multiplicative inverse of $Z$. – xorhash May 15 '20 at 18:23