Geometric interpretation of an Edwards curve
The geometric interpretation of the "addition law" on Edward Curves is not the same as for Weierstrass Curves.

The correct interpretation for this kind of curves is "adding their angles".
It works as on a clock. Of course, as for the Weierstrass curves, the geometric interpretation stands for the curve over the real numbers and not in a finite field (useful for cryptography).

You can give a look to the ECCHacks: a gentle introduction to elliptic-curve cryptography (starting at page 6) by Daniel J. Bernstein and Tanja Lange