What happens if you multiply a point with its order on complete Edwards curves? I was recently working with some ECC crypto and stumbled across the following phrase on the SafeCurves page:


  The rational points of a complete Edwards curve are the pairs (x,y) of
  elements of F_p satisfying the equation; there is no extra "point at
  infinity".


Normally, if you have a point $G$ on a curve $E$ with order (of the point) $q$, multiplying $qG=0G=\mathcal O$ results in the point at infinity.

But apparently complete Edwards curves don't have said extra point.

So what happens in this case?
Actually multiplying an elliptic curve point $G$ by its order $q$ gets you to the neutral element of the group, i.e., the point $O$ such that $P+O=P$ for any point P.

In a generic elliptic curves this point is the point at infinity $O$. However on Edwards curves, this point is $(0,1)$, i.e., it can be represented by affine coordinates. This is also stated on wikipedia. 

Therefore, multiplying a point by its order, on an Edwards curve, gets you the point $(0,1)$

This means that on Edwards curves: $P+(0,1)=P$ for any point $P$.