It’s been a while since we’ve looked at a new type of geometry, so today I’d like to introduce inversive geometry. Let’s look at a few pictures first for some motivation.
We would like to consider the operation of inverting points, as illustrated above. Suppose the red circle is a unit circle, the white dot is the origin, and consider the blue point P. Now define P′ (the orange dot) to be the inverse point of P as follows: draw a ray from the origin through P, and let P′ be that point on the ray such that
where denotes the distance from A to B. That is, when you multiply the distances from the origin of a point and its inverse point, you get 1.
Now why would you want to do this? It turns out this operation is actually very interesting, and has many unexpected consequences. Consider the circles in the figure below.
Start with the blue circle. Now for every point P on the blue circle, find its inverse point P′, and plot it in orange. The result is actually another circle! Yes, it works out exactly, although we won’t prove that here. Note that since every point on the blue circle is outside the unit circle (that is, greater than a distance of 1 from the origin), every inverse point on the orange circle must be inside the unit circle (that is, less than a distance of 1 from the origin).
This is just one geometrical property of inversion, and there are many others. I just want to suggest that the operation of inversion has many interesting properties.
If you pause a moment to think about this operation, you’ll notice there’s a little snag. Not every point has an inverse point. There’s just one point which is problematic here: the origin. According to the definition,
But the distance from the origin to itself is just 0, and it is not possible to multiply 0 by another number and get 1, since 0 times any real number is still 0.
Unless…could we somehow make the distance from O to O′ infinite? It should be clear that O′ cannot be a point in the plane different from the origin, since any point in the plane has a finite distance to the origin (just use the Pythagorean theorem).
So how do we solve this problem? We add another point to the usual Euclidean plane, called the point at infinity, which is usually denoted by the Greek letter ω.
You might be thinking, “Hey, wait a minute! You can’t just add a point because you need one. Where would you put it? The plane already extends out to infinity as it is!”
In a sense, that’s correct. But remember the idea of this thread — we’re exploring what geometry is. When we looked at taxicab geometry, we just changed the distance, not the points. And with spherical geometry, we looked at a very familiar geometrical object: the surface of a sphere. We can also change the points in our geometrical space.
How can we do this? Though perhaps a simplification, we must do this consistently as long as it is interesting.
What does this mean? In some sense, I can mathematically define lots of things. Let’s suppose I want to define a new operation on numbers, like this:
Wow, we’ve just created something likely no one has thought of before! And yes, we probably have — and for good reason. This operation is not very interesting at all — no nice properties like commutativity or associativity, no applications that I can think of (what are you going to do with the 37?). But it is a perfectly legitimate arithmetical operation, defined for all real numbers a and b.
As I tried to suggest earlier, the operation of inversion is actually quite interesting. So it would be very nice to have a definition which allowed every point to have an inverse point.
But if we want to add ω, we must do so consistently. That is, the properties of ω cannot result in any contradictory statements or results.
It turns out that this is in fact possible — thought we do have to be careful. Although we can’t go into all the details of adding the point at infinity to the plane, here are some important properties that ω has:
- The distance from ω to any other point in the plane is infinite;
- ω lies on any unbounded curve (like a line or a parabola, but not a circle, for example);
- The inverse point of ω is the origin; that is, O′ = ω and ω′ = O.
So we can create an entirely consistent system of geometry which contains all the points in the Euclidean plane plus a point at infinity which is infinitely far away from every other point. We usually call the Euclidean plane with ω added the extended plane.
With the new point, we can now say that every point in the extended plane has a unique inverse point.
This idea of adding a point at infinity is also important is other areas of mathematics. In considering the complex plane — that is, the set of points of the form a + bi, where i is a solution of the equation — it is often useful to add the number ∞, so that division by 0 is now possible. This results in the extended complex plane, and is very important in the study of complex analysis.
The point at infinity is also important in defining stereographic projection. Here, a sphere is placed on the plane, so that the South pole is the origin of the plane. The sphere is then projected onto the plane as follows: for any point on the sphere, draw a ray from the North pole through that point, and see where it intersects the plane.
Where does the North pole get mapped to under this projection? Take one guess: the point at infinity!
So here is another new geometry! As we get introduced to more and more various geometrical systems, I hope you will continue to deepen your intuition about the question, What is a geometry?