point addition for elliptical curves
Addition of a point with itself is "just" the limit case of normal addition. Geometrically, point addition starts with a line that goes through the two points to add; point doubling uses the tangent to the curve on that point, which is what you expect to have when the two points are made very close to each other.

In that geometric interpretation, the tangent to the curve on a point $P = (x,0)$ is vertical, and thus won't intersect the curve anywhere else; or, more accurately (using projective geometry terminology), that vertical line meets the curve again only at infinity. The sum of $(x,0)$ with itself is thus the point at infinity. This also means that $(x,0)$ is its own opposite.

Now, in cryptography, we use curves in finite fields, which are discrete and don't look like curves. One interesting point to make is that a point $P = (x,0)$ is such that $2P = \mathcal{0}$, i.e. it is a point of order 2. Since an elliptic curve is a group, if there is a point of order 2, then the total curve order must be a multiple of 2. Many standard curves (e.g. the very widespread NIST curve P-256) have been chosen to have a prime order, hence odd; it follows that, for such curves, there is no point $(x,0)$, and the question of doubling such a point does not arise at all.