Context: this was left on the cutting floor for my Algebra (intersects) Geometry session handout. Mostly an amalgamation of shower thoughts.
I’ve peppered this with many thought exercises (“TE”), so feel free to think along with me (:
A running theme in these sessions will be that we’re trying to enforce configurational symmetry, and this is perhaps better understood with the following slogan:
Properties should not depend on configuration.
The most basic examples was back in the very first section, where we decided that any pair of lines should really intersect at one point, even parallel ones. One way to enforce this is to move to the real projective space by adding points at infinity that correspond to “directions”.
Here we want to do something similar, but now with circles. A pair of intersecting circles must intersect at two points. Or must they?
Consider the following setup: two circles $\gamma,\omega$ with their centers on $\ell$ and their intersections on some $\ell' \perp \ell$.
Initially, perhaps they have two distinct intersections. As you slowly pull their centers away, the two intersection points get closer and closer, until finally they overlap. Aha! Now there’s only one intersection point (“$\gamma$ and $\omega$ are tangent”), but since we’re been watching the whole situation closely since the start we know that it’s secretly two intersection points.
It is at this point the craziest thing happens: if you pull the circles apart just a little bit more, then both intersection points vanish! Where did they go?
Exercise. Let’s look for them. Let $\ell,\ell'$ be the $x$ and $y$-axes, and let the circles have radii $r_1$ and $r_2$, and perhaps let $\gamma$ be centered at $(t,0)$. Where are the two intersection points, and where do they go?
The shocking conclusion is that there are two intersection points, just not in $\mathbb R^2$. Two apparently non-intersecting circles do intersect, in $\mathbb C^2$.
TE. Do two circles in $\mathbb C^2$ necessarily intersect at two points (up to multiplicity)?
TE. Implicit in the above TE was the way we’re going to define distance. What is it, and is it a weird choice?
A very natural manifestation that you should care about is as follows: there is something special about three circles sharing the same two intersection points. (For instance, they have a common radical axis.)
But now we do the process described above and pull the common intersection points out of existence. The strange thing is that somehow, even though we don’t see it any more, the circles do still have two common intersections, through which we can draw a line.
And that line almost has to be the radical axis, right? Going back to the geometry we know, we say that three circles are coaxal (“co”-radical-“axal”) if they share a radical axis, and if they intersect this usually means that both intersection points are in common. But by our expectations of configurational symmetry, three coaxal circles must actually have two common intersections somewhere…
Exercise. Check, using the above coordinates, that three coaxal circles do intersect somewhere.
In some sense, we see again that it really doesn’t matter if circles intersect or not, and coaxal circles are a “first example” where there is some tangible property shared across seemingly “distinct” cases.
(Unresolved) TE. I’ve always wondered about how you can invert two circles into concentric circles. This is easy enough for non-intersecting circles, but somehow not quite the same for intersecting ones. One possible alternative is to invert them onto orthogonal circles, but that isn’t quite as satisfying…
Here’s a fun problem if you’ve never heard of this:
(some aops contest) Let $\omega_1,\omega_2$ be two fixed circles, $P$ be a variable point. $PA_i,PB_i$ are tangents to $\omega_i$, and $A_1B_1,A_2B_2$ intersect at $Q$. Show that regardless of where $P$ is, the midpoint of $PQ$ will vary along a fixed line.
(ISL 09? G3)
Essentially, points are a limiting case when a circle has zero radius, and you can do the usual things we can do with circles (notably, power calculations).
Now what?
edit from the future: Glen has fulfilled a wish of mine by writing this article. He’s managed to pen down a superset of everything I had in my head for this section.
edit from the future: this is not to be confused with the blowup from algebraic geometry.
This is an incomplete idea, based on stuff I saw in this paper.
We see Hart’s theorem, which is a trick where under some circumstances we can abuse inversion and duality to turn three lines into three arbitrary circles.
The paper offers a crazy generalization, in which all points seem to have turned into inscribed circles.
Here’s a less crazy example: generalized Monge. We see we can indeed replace all triples of intersection with inscribed circles.
I suspect the mechanism for this is as follows: in the $(a,b,ir)$ interpretation, points are just points with $r=0$. But with some suitable “rotation”, we can usually generalize facts in the space to any “general configuration”.
An example: we know easily we can translate things up and down without breaking, and this translates to increasing the radius of things by a fixed amount. This trick is sometimes called “dilatation”.
The most interesting part for me is that this crazy general theorem instant-kills Emelyanov’s theorem.
If you think about it, it’s also dim 2 over $R$, so it’s some kind of surface. What surface is it?
One way to do it is to grind coordinates… what happens? (I thought that it corresponded to $$(\Re z_1)^2 + (\Re z_2)^2 = 1/(1-t^2)$$ for some real parameter $t$, but on further thought this starts to not make sense, because everything has a square root, right?)
One weird thing I realized is that point circles aren’t point circles anymore in $\mathbb C^2$. Hmm.