The Projective Plane: A Visual Introduction

The statement "any two distinct lines intersect in a point" is almost true in normal plane geometry. The exception, of course, is the case of two parallel lines. However, from real experience we know from the rules of perspective that two parallel lines "converge" very far away, even if we know that they in fact maintain the same distance apart.

From this, we naturally comes the intuition that "parallel lines intersect at infinity." Certainly this tidies up our intersection statement because it provides a way for even parallel lines to intersect. But what does "at infinity" mean? Is there really a "point" there? The notion of projective space makes these ideas explicit and rigorous.

We focus on the (real) projective plane, the extension of the normal plane to include these "points at infinity" where parallel lines intersect. The set of points in the projective plane is defined, somewhat enigmatically, as "the set of lines through the origin in three-dimensional space." Defining each point to be a line in a different space seems extremely confusing at first, but there are multiple ways to visualize this concept.

The first method of visualization illustrates how the projective plane is related to the ordinary plane (sometimes called the affine plane). Consider three-dimensional space with ordinary coordinates x,y, and z as shown. The plane labeled z = 1 contains all points for which the z coordinate is 1, namely all those of the form (x,y,1). Clearly this plane is just like the ordinary two-dimensional plane (under the correspondence (x,y,1) → (x,y)), only embedded in three dimensions, like a flat sheet of paper in our world (but infinite). The dotted line shown that passes through the origin intersects the plane at the particular point (a,b,1). Remembering that the projective plane is meant to be an extension of the affine plane, we identify the dotted line with the point where it intersects the plane. Clearly, for each point in the z = 1 plane, there exists exactly one line through the origin and the given point. This shows how the ordinary plane is a subset of projective space (the set of lines through the origin)!

However, not every line through the origin intersects the plane z = 1. For instance, the x-axis, the y-axis (both shown), and any other line in the plane of these two axes only contain points for which the z coordinate is 0 and can never intersect the plane z = 1 (to be clear, the three-dimensional space considered here does not have its own points at infinity!). Therefore, these lines cannot correspond to points on the ordinary plane. These special lines, in fact, are the points at infinity in the projective plane.

The above visualization illustrates the connection between the projective plane and the affine plane. It also indicates that there are many points at infinity, one for each line through the origin "lying flat" in the xy-plane. However, it fails to indicate how points at infinity are truly the intersections of parallel lines. For this, we use another visualization that chooses different representative points.

Using a sphere (or a hemisphere, to be more precise) to represent the projective plane is just as legitimate as using a plane: all that matters is that there is one point for each line through the origin. It does not matter which points we choose.

In fact, nearly every person is intimately familiar with this representation of projective space! Imagine that it is a clear night and you go out to look at the stars. You catch sight of the familiar constellation Orion, the hunter. The stars marking Orion's shoulders are Betelgeuse and Bellatrix, which we perceive to be neighboring stars that connect to form the figure of Orion. In fact, however, Betelgeuse is between two and three times as distant as Bellatrix. When we look up at the sky, we do not perceive the true three-dimensional space but points of light etched into the inner surface of the celestial sphere passing overhead. Stars in similar directions, regardless of their distances, are projected onto nearby points. This is why the result of treating all points along a line through the origin as equivalent is known as projective space.

It is clear, however, that every line through the origin intersects the sphere at exactly two points, while there can only be one representative for a point of projective space. Thus, by convention we consider only intersections with the upper hemisphere (just as in our example of the night sky - one cannot see stars looking downward!). This leaves only the "horizontal" lines intersecting the equator of the sphere twice. For these, we choose the points of intersection for positive y-values (the area colored dark green above) and finally the x-axis is represented by the dark red point of positive x. The projective plane is therefore the union of the yellow upper hemisphere, the dark green semicircle, and the dark red point. The latter two parts are the points at infinity.

The above image shows how the affine plane (and our first visualization) relate to our second visualization of the projective plane as (part of) a sphere. Lines through the origin (O) and a point in the upper hemisphere intersect the plane to form a one-to-one correspondence. As we would expect, points at infinity correspond to lines through the sphere's equator that are parallel to the plane and are therefore not part of our original affine plane.

Finally, the sphere illustrates how the projective plane solves the motivating problem of parallel lines than began this post.

Two parallel lines in the plane correspond to precisely the same lines in our first visualization, which indeed embeds a "copy" of the affine plane in three-dimensional space. When these parallel line are transferred to the sphere in the same manner that the point was above (remember: each transferred point represents a line through the origin and "transferring" a point is merely choosing a different representative), the figure above is the result. However, it is evident that the resulting arcs on the sphere intersect at the equator (green circle) and we know the equator contains the points at infinity! Though there appear to be two intersections, recall that points diametrically opposite from one another are on the same line through the center, so that these points are identified as one in the projective plane. We have our desired result: two parallel lines intersect in exactly one point.

The next post provides an algebraic description of the projective plane and explores more of its properties.

Sources: Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton, https://www.math.toronto.edu/mathnet/questionCorner/qc_hlimgs1/image87.gif, http://jwilson.coe.uga.edu/EMAT6680Fa11/Chun/Final1/4.png, http://courses.cs.washington.edu/courses/cse557/98wi/readings/xforms/diagram/homogeneous.gif, http://earthsky.org/astronomy-essentials/how-far-is-betelgeusehttp://en.wikipedia.org/wiki/Projective_space