# Doubling Points - 2P, 4P, 8P, etc.

Based on the curve y^2 = x^3 - 2x + 4, the tangent line to the point P is equivalent to a "double root" of a cubic, which then crosses the curve at R. We take -R = 2P, and find the tangent line to that point, which then crosses the curve at the point S. We take -S = 2R, and find the tangent line to that point, which then crosses the curve at the point T. etc. This effectively "doubles" P each time.

Doubling a point P on an elliptic curve to find 2P, and then doubling 2P to find 4P, involves both geometric intuition and algebraic manipulation under the elliptic curve's group law. Here's how the process works, both geometrically and visually:

Geometric Description of Doubling P to Find 2P

1. Identify the Point P: Start with a point P on the elliptic curve. The point P has coordinates (x_1, y_1).

2. Tangent Line at P: Draw a tangent line to the elliptic curve at the point P. This line will touch the curve only at P if P is not an inflection point.

3. Intersection with the Curve: The tangent line will generally intersect the elliptic curve at exactly one other point. Because of the symmetric property of the curve about the x-axis, this intersection point is technically the point -2P, which is the reflection of 2P across the x-axis.

4. Reflect to Find 2P: Reflect the intersection point across the x-axis to find 2P. The point 2P is directly below (or above) the intersection point, depending on the curve's orientation.

Doubling 2P to Find 4P: To double 2P to find 4P, repeat the process using 2P as the starting point.

1. Tangent Line at 2P: Draw a tangent line to the elliptic curve at 2P.

2. Intersection with the Curve: This tangent line will intersect the curve at another point, which is -4P, the reflection of 4P.

3. Reflect to Find 4P: Reflect the intersection point across the x-axis to find 4P.