What exactly is the "point at infinity"?
It's a point that is added to the points on the curve. Together they form a group. It has the following properties:
- (x,y) + (x,-y) = infinity
- (x,y) + infinity = (x,y)
- infinity + (x,y) = (x y)
In other words, the point at infinity is the identity element of the addition in the group. Therefore some people write the point at infinity as "0".
Is there more than one "point at infinity"
How can I identify if my EC generated x and y are the "point of infinity"?
The point at infinity is not on the curve, so it does not have x or y coordinates. It appears whenever a point is added to its own negation (for which the normal addition rule has no answer).
Is there a way to calculate the "point of infinity"?
It simply is "the point at infinity", there is nothing to be calculated about it.