Infinity Point on an Elliptic Curve

This section describes how the infinity point is used to represent the intersection of vertical lines and elliptic curves.

In order to completely define the addition operation on an elliptic curve, we need to introduce a special point on the curve, the infinity point.

Consider the following addition operation of P and Q on an elliptic curve, where Q is the symmetrical point of P on the same curve. Or we can express Q as:

```Q = -P
```

In this case, we can follow our geometrical algorithm to find the R = P + Q:

1. Draw a straight line passing P and -P. This will be a vertical line.

2. Mark the third intersection of the vertical line and the elliptic curve as -R, which will be the infinity point of the 2-dimensional space. This is where the vertical line and the elliptic curve will eventually intersect. In other words, -R is the infinity point:

```-R = (∞, ∞)
```

3. Mark the symmetrical point of -R on the other side of the x-axis of the elliptic curve as R which will be the same infinity point. This is because the infinity point has no difference between positive or negative signs. In other words, R is the infinity point:

```R = (∞, -∞) = (∞, ∞)
```

Here is a diagram that illustrates how a vertical line intersects with an elliptic curve at the infinity point.