The more I thought about it, the more I realized I just couldn’t stop at just one post on inversive geometry. There is so much interesting geometry to discuss….
This image of a cardioid, made up of a collection of tangent circles, is one of my favorite geometrical constructions. It’s created by beginning with a base circle (in black). Choose any point on the circle (like the one in black), which we take to be the origin.
Now for any other point on the black circle (like the point in red), draw a circle with that point as center, and the radius as the distance from the origin, as in the figure. If you do this several times, the circles you draw will envelope a cardioid.
Why do you get a cardioid? It’s actually not too difficult to prove using inversive geometry. We won’t go through all the details, but you’ll see enough to get the gist of the proof.
So let’s go! There’s actually a lot going on once you’ve added the point at infinity, ω. If you’ve forgotten about ω, you might want to look at Day091 to refresh your memory. One property of ω which will be important to our discussion today is that ω lies on any curve which is unbounded. And the simplest unbounded curve is just a line.
So imagine a line, and some point P on that line. Now chose a direction long the line, and think about moving along the line in that direction. What happens?
Well, eventually, you’ll start moving further and further away from the origin. And as you keep moving further from the origin, you’re actually moving “closer” to the point at infinity. Not closer in a numerical sense of the distance from ω, since you are always infinitely far away from ω. But closer in the sense that the line “ends” at the point at infinity. If ω lies on the line, you’ll need to able to get there — even it takes forever….
Yeah, a bit to wrap your head around. But it gets even better! Think back to where you started — the point P — and then move along the other direction along the line. Keep going…as you approach the point at infinity from the other side of the line.
Do you see what’s happening? Both ends of the line actually meet at ω! Yes, the line is a closed curve in inversive geometry, like a circle is in Euclidean geometry.
We’ll come back to this point in a moment, but next we want to ask the following question: what is the inverse curve of a line? Recall that the inverse curve of a line is that curve obtained by taking all the individual points on a line, and then taking all of their inverse points.
Interestingly, the inverse curve turns out to be a circle — and not just any circle, but a circle which goes through the origin. As the proof isn’t all that difficult, we’ll take a few moments to work through it.
So suppose the red circle centered at O has radius 1, and choose a point P. Now draw the line through P such that the segment OP is perpendicular to the line, and consider another point Q on this line. Thus, triangle OPQ has a right angle at P.
Since the point at infinity, ω, is on the line, its inverse point, ω′ = O, must lie on the inverse curve. Thus, the inverse curve (shown in blue) must pass through the origin. Of course P′ and Q′, the inverse points of P and Q, must also lie on the inverse curve.
Then by the property of inverse points, and so
But because inverse points lie on the same ray through the origin, then and must have the same measure, making triangles and similar triangles.
What does this mean? This implies that is also a right angle. To summarize: given any point Q on the line, its inverse point Q′ has the property that when joined to O and P′, a right triangle is formed with a right angle at Q′.
But because any triangle inscribed in a semicircle is a right triangle, this means that the set of all inverse points Q′ forms a circle with diameter OP′. Neat!
Thinking back to ω being on the line, we see that as we begin at P and move either direction along the line toward the point at infinity, here’s what we’re doing on the inverse curve: we’re starting at the point P′ and moving either direction around the circle toward the origin. It really does make sound geometrical sense.
Will this work for any line? If you think about it for a moment, you’ll find that it does…almost. Our geometrical argument fails if the line actually goes through the origin. But it shouldn’t be difficult to convince yourself that the inverse curve of a line going through the origin is actually the same line! The two points where the line intersects the circle of inversion stay in the same place, but the other points switch in pairs.
Where are we going with all this? It turns out that adding the point at infinity does, in some sense, make lines behave like circles — we can even imagine a line as a “circle with infinite radius.” And where is the center of this circle of infinite radius? The point at infinity, of course….
As mentioned last week, the inverse curves of circles not passing through the origin are also circles not passing through the origin. To summarize:
- Inverses of circles not passing through the origin are also circles not passing through the origin;
- Inverses of circles passing through the origin are lines not going through the origin (and vice versa);
- Inverses of lines through the origin are also lines through the origin, and each such line is its own inverse.
Now here’s the next step. Given the close relationship between lines and circles in inversive geometry, we define a Circle in inversive geometry (capital “C”, like we did in spherical geometry) to be either a line or circle. Then here’s the revised summary:
- Inverses of Circles are Circles.
Beautifully simple. The ability to make simple statements like this with real geometrical meaning is one more reason why inversive geometry it so very interesting.
And with that, we’ll have to call it a week…. In the next (and final) post on inversive geometry, we’ll briefly look at the algebra of inverse curves, and finish the cardioid construction mentioned above. Until then!
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