Addition Operation on an Elliptic Curve

This section describes the addition operation on an elliptic curve geometrically. The addition of points P and Q on an elliptic curve is a point R on the curve, which is the symmetrical point of -R, which is the third intersection of the curve and the straight line passing through P and Q.

The addition operation on an elliptic curve is defined geometrically as below:

For any given two points P and Q on an elliptic curve, the addition operation of P and Q will result a third point R on the curve by running the following geometrical algorithm:

1. Draw a straight line passing P and Q

2. Mark the third 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.

The above geometrical algorithm is also called "rule of the chord".

If we use the plus sign "+" as the addition operator, the addition operation of points P and Q on an elliptic curve can be expressed as:

```P + Q = R
```

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