Introducing the projective plane
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When I got the idea of creating visuals involving the sphere, my mind quickly leapt to the idea of visualizing the projective plane. It is an important space that mathematicians are constantly referring back to, but the space was always a bit mysterious to me, and I wanted to take the opportunity to clear things up. The projective plane can be an unfamiliar object to some so let's discuss it a bit.
The projective plane is just like the plane that we're all familiar with, except that for every direction in the plane, we add a point to the plane called a point at infinity. Unlike with stereographic projection, we have a point at infinity for every direction in the plane, not just one. Every line in the plane contains the point at infinity corresponding to its direction. We also add an additional line that is composed of all the points at infinity. This space formally captures the observation that groups of parallel lines in perspective drawing all seem to converge at a single point.
The axiom that set projective geometry apart is known as the elliptic parallel axiom. Two distinct lines must intersect at exactly one point. In the Euclidean plane geometry that we're all familiar with, we know that this is not true for parallel lines. However, in projective geometry, lines all contain a point at infinity corresponding to their orientation. Since parallel lines have the same orientation, they all intersect at their point at infinity, just like the parallel lines in perspective drawing.
The projective plane is an example of a projective geometry, which is a simplified form of geometry, satisfying a certain set of axioms, that keeps the notion of lines and points but removes all of the others, such as distance and angles. So, in order to define a projective geometry, all you need to do is define what the lines and points are, and guarantee that they satisfy the list of axioms.
This stripped down notion of geometry allows us to view the sphere as a projective geometry whose "lines" are the great circles of the sphere, and whose "points" are antipodal pairs of points on the sphere. We can do this because the only thing that matters in projective geometry is that we have some definition of lines and points and we don't care about other geometric notions such as the straightness of those lines.
As a matter of fact this projective geometry is equivalent to the projective plane, and this equivalence is visualized in the two visualizations below. The mapping from pairs of antipodal points to the plane is \[ (x,y,z) \mapsto (x/z, y/z, 0) \] for points not on the equator. On the equator, the formula is invalid and does not apply, since \( z = 0\). Instead, on the equator, we map the antipodal points to the point at infinity corresponding to the slope of the line from the origin to \( (x,y) \). You can see from the formula that the mapping sends the point \( (x,y,z) \) to the same value as its antipodal point \( (-x,-y,-z) \). That's why we need to identify antipodal points with each other, so as to make this mapping one-to-one.
This procedure of identifying certain points of a space together to create a new space is a common and useful construction in mathematics. It is called taking a quotient of the space and the resulting space is called a quotient space.
The visualization above displays this mapping for individual points, and the visualization below displays the mapping for lines. Recall that the "lines" on the sphere actually look like circles. Again, in projective geometry, the only aspect of lines that we take into consideration is what points they contain. By interacting with the visualization, you can see that points that are further out from the origin correspond to points on the sphere closer to the equator. Again, the equator of the sphere itself maps to the points at infinity.
From the point of view of projective geometry, the two versions of the projective plane are completely equivalent. However, for the purpose of analysis, the spherical version is generally easier to work with because all of its points and lines exist on an equal footing. The planar version can be awkward to work with because it has a special class of points and a special line that lie at infinity that have be examined separately from the regular points.