The last type of geometry I discussed was inversive geometry, which is obtained by adding a point at infinity to the Euclidean plane. Recall that as long as we had a consistent, useful model of this extended plane, it made perfect sense to define this new type of geometry.
Today, we’re going to add a line at infinity — creating what is called a projective geometry. There are in fact many different types of projective geometries, but let’s just try to understand one at a time….
You might remember one important property of ω, the point at infinity in inversive geometry: it was on every unbounded curve, and in particular, on every line. We need to be a bit more specific with our projective geometry, in the following sense. Every line will intersect the line at infinity — denoted by λ — but not every line will intersect at the same point.
Consider all lines with some given slope m. We then add one point to λ which lies on these lines, but no other lines. In other words, each point on λ corresponds to an infinite family of parallel Euclidean lines — since now, with the addition of λ, there are no parallel lines in the projective plane. The point at which two parallel (in the Euclidean sense) lines intersect is determined by their slope.
How does this differ from inversive geometry? Well, in inversive geometry, if two lines had different slopes, they intersected in two points: the usual finite point of intersection you learned about in algebra class, as well as ω. But in projective geometry, two lines with different slopes intersect in only one point, since they intersect the line at infinity in two different points.
What this means is that any two distinct lines always intersect in exactly one point. See the difference? Parallel lines (in the Euclidean sense) intersect in a point on λ, and nonparallel (in the Euclidean sense) lines intersect in the same point they did in Euclidean geometry.
Now let’s look at the dual question (recall the discussion of duality, an important concept in spherical geometry): what about a line between two points? Two finite points generate a line, as usual. If one point is finite and one lies on λ, the line generated is that line through the finite point with the slope corresponding to the point on λ. And if both points are infinite, then the line through them is just λ.
Thus we have the following duality: two distinct lines determine a unique point, and two distinct points determine a unique line. Again, duality is an extremely powerful concept in geometry, so the fact that points and lines are dual concepts really is a legitimate justification for thinking about projective geometry.
Projective plane geometry is a broad subject — but some of my favorite objects to look at in projective geometry are the conic sections: ellipses, parabolas, and hyperbolas. This is because from the point of view of projective geometry, they are, in a sense which we’ll look at right now, all the same.
This sounds odd at first reading, but follow along. The first question we need to consider is what points on λ lie on unbounded curves. For lines, it’s easy — it’s just the point on λ corresponding to the slope of the line.
Now what about a parabola?
You can see from the image that as points on the parabola move further away from its vertex, the tangents have slopes of ever-increasing magnitude. Determining the precise slopes is an easy exercise in calculus — but what is important is that the tangent lines approach, in slope, the axis of the parabola. In this case the axis is vertical, but of course the parabola may be rotated.
What this means is that the parabola intersects the line at infinity in the point corresponding to the slope of its axis. Again, the tangent lines on either side approach this slope, but from two directions — from above and from below.
Now hold this thought while we look at a hyperbola.
Let’s see how we travel along the hyperbola. As we move to the upper right toward the open red circle, we are moving closer to the red dashed asymptote — meaning we approach the point on λ corresponding to the slope of the red asymptote. But here is where it becomes interesting: as we cross the line at infinity, we come back along the asymptote toward the filled red circle on the other branch of the hyperbola!
Then we continue moving along the left branch, getting continually closer to the blue asymptote, approaching that point on λ corresponding to the slope of the blue asymptote as we pass through the blue filled circle. Then we cross the line at infinity again, and jump to the right branch of the hyperbola toward the open blue circle.
Can you see what this means? In projective geometry, a hyperbola does not have two separate branches. Adding the line at infinity allows us a means to jump between the two branches.
Let’s look at these scenarios from a slightly different perspective. Suppose that somehow, we can actually draw the line at infinity. It’s the blue dashed line in the figure below. Easy.
Can you see where we’re going with this? The left circle represents an ellipse, since it is a bounded curve and cannot intersect λ. The middle circle represents a parabola — it’s a curve which intersects the line at infinity at exactly one point, and so is actually tangent to λ. The right circle represents a hyperbola, since it intersects the line at infinity at two points. It may look strange from this perspective, but this right circle really does represent the hyperbola illustrated above.
To summarize: if a conic does not intersect the line at infinity, we call it a Euclidean ellipse, if it is tangent to λ, we call it a Euclidean parabola, and if it intersects λ in two points, we call it a Euclidean hyperbola.
Because you see, in projective geometry, there is no distinguished line. Every line is just like any other. So, in some sense, these is just one conic section in projective geometry!
What we have looked at today is what would be called an embedding of the Euclidean plane in the projective plane. We saw how we could just add a line at infinity with certain properties, and our plane would behave “projectively.” But from the perspective of projective geometry, you’ve just got conic sections and lines, with possibly 0, 1, or 2 intersection points as illustrated above. A little mind-blowing, but absolutely true.
Of course we’ve only been able to scratch the surface of this really amazing geometrical world today. Hopefully you will be inspired to learn a little bit more about it….