The Projective Plane

The projective plane plays a critical role in the study of elliptic curves, providing a broader framework that extends the usual Euclidean ("affine") plane to include points at infinity. This extension is crucial for defining the group structure of elliptic curves, particularly for handling operations that involve the 'point at infinity', which serves as the identity element in the group. By considering elliptic curves in the context of the projective plane, we can ensure that the curve's group law is well-defined and complete for all points (Washington, L. C., 2008, pp18-20, Section 2.3).

Below is an animation of some of the properties of the projective plane:

This animation indicates:

1. The green plane, called the "affine plane", at z = 1, represents the 2-dimensional plane on which we would normally draw our lines, or curves.

2. The orange point, P, on the affine plane (the green plane at z = 1), which is moving along the red line: x/2 + y - 1/2 = 0y = z - 1

3. The fact that, as the line (orange) connecting P to the origin of our 3-dimensional space, traverses the space it marks out a plane (blue) represented by: x + 2y = z

4. Furthermore, if we were to keep extending P infinitely along the path along which it is travelling (in either direction) , the orange line, attached to the origin, would get closer and closer to being on the z = 0 plane (marked as the transparent grid in the animation). We will explore this further below.

To Infinity, and beyond

To emphasise point 3 above, the next animation is a "zoomed out" version of the previous animation (see the scale indicated on the z-axis - the animation above has the z=1 plane near the top - this animation below extends to z=20, and upper red line is still at z=1 ).

Furthermore, the green plane at z = 1 has been removed, to create a clearer view of:

1. the point P (the orange point)

2. the line connecting it to the origin (the orange line), and

3. the lines parallel to the xy-plane (the red lines at z = 1, and z = 0) on the plane marked out by the line connecting P, to the origin, as it moves with P.

The aim here is to imagine the point P going "to infinity", to visualise the fact that the line connecting P to the origin gets closer and closer to the z = 0 plane - eventually, at the limit, becoming the red line, on z = 0.

This line, on the z = 0 projective plane, are the "points at infinity" defined for P, as it goes to infinity, along the line indicated by the line at z = 1. While the "point at infinity" is often referred to as a single point (in this case it would be [ -2, 1, 0 ] ), it is actually the infinite number of points on the line, on the z = 0 plane, parallel to the line on the affine plane, passing through the origin.

Parallel Lines Meet At Infinity (on the same projective line)

The next animation emphasises the idea that any parallel line on the affine plane meets all other parallel lines, at the the limit, at the same "points at infinity", on the projective plane (Washington, L. C., 2008, pp18-20, Section 2.3).

In the animation below we see 4 lines, all created by various points moving on the z = 1 affine plane:

1. the original line (orange) created by a point travelling along the line y = -x/2 + 1/2 on the z = 1 plane

2. a second line (purple) created by a point travelling along the line y = -x/2 + 2 on the z = 1 plane

3. a third line (green) created by a point travelling along the line y = -x/2 + 4 on the z = 1 plane

4. a fourth line (grey) created by a point travelling along the line y = -x/2 + 8 on the z = 1 plane

At the beginning of this animation, the space is oriented to emphasise the fact that each of the 4 lines are on the affine (z = 1) plane. It then switches to a different perspective, that helps to visualise the relative positions of those lines, and the fact that they are parallel to each other in 3-dimensional space, but also, specifically, when viewed "above" from the perspective of the x-y plane.