Same Point Addition on an Elliptic Curve

This section describes how to perform the addition operation of a point P to the same point P on an elliptic curve. In this case, we will draw a straight line that passes P and tangent to the curve to find -R.

What happens if we want add a point P on an elliptic curve to itself?

In this case, we need to modify our geometrical algorithm as the following to find R:

1. Draw a straight line passing P and tangent to the elliptic curve.

2. Mark the second intersection of the straight line and the elliptic curve as -R.

3. Mark the symmetrical point of -R on the other side of the x-axis of the elliptic curve as R.

If we use the plus sign "+" as the addition operator, the addition operation of a point P to itself on an elliptic curve can be expressed as:

```P + P = R
```

The above expression can abbreviated as the following, if we follow algebraic formula convention:

```P + P = 2P = R
```

Here is a diagram that illustrates how to perform the addition operation on an elliptic curve geometrically (source: hackernoon.com):