Elliptic curve cryptography G*G I understand how ECC works for the point multiplications and stuff. 
All we normally do is to multiply a scalar number (lets say d as a private key) with the base point generator G to get a public key point on an elliptic curve. 
I am trying to tweak the algorithm for some other purposes so I am just wondering whether an operation like G*G which is a point multiply another point is possible or not. And if it is possible how can we calculate that?
Such an operation is possible.

The ways of calculating that appear to be different for each curve.