Point Addition

Adding Points

We can “add” two points on the curve to get a third point.

This isn’t regular addition. It’s a geometric operation.

How It Works: P + Q

To add two different points $P$ and $Q$:

Step 1: Draw a straight line through $P$ and $Q$.

Step 2: This line hits the curve at exactly one more point. Call it $R'$.

Step 3: Reflect $R'$ across the x-axis. This is $R = P + Q$.

Why Does the Line Hit a Third Point?

The curve equation is cubic (has $x^3$).

A line can intersect a cubic curve at most 3 times.

If the line already passes through $P$ and $Q$, there must be a third intersection point.

Point Doubling: P + P

What if we want to add a point to itself?

We can’t draw a line through just one point. So we use the tangent line instead.

How It Works: 2P

To compute $P + P = 2P$:

Step 1: Draw the tangent line at $P$ (the line that just touches the curve).

Step 2: This tangent hits the curve at one other point $R'$.

Step 3: Reflect $R'$ across the x-axis. This is $2P$.

The exact angle of the tangent comes from differentiating the curve equation. The math works out cleanly, but we won’t go into it here.