#apollonius

[qqquinoxtikk]

Someone should make a ballet based on Historia Apollonii Regis Tyris (Apollonius Prince of Tyre), which Shakespeare basically copied for his play Pericles, Prince of Tyre. The drama. The characters. The fairy-tale. *Mwah*

[katerinaaqu]

What people think Medea was like at first in the Argonautica:

What Medea truly was like at first in the Argonautica:

[katerinaaqu]

The third book of Argonautica will never stop having a really funny beginning to me! Like I so love it that Athena and Hera both talk on actually HOW can the Argonauts actually get the Golden Fleece and the conversation more or less goes like:

Hera: Alright Athena, our heroes need to actually take the thing and the path is hard. Any ideas? Athena: I honestly don't know. I myself am wondering. Got ideas but none seem efficient... Hera: Alright then I'll tell you what, we go to your sister and tell her order her son shoot that witch Medea with his arrows so she will fall helplessly in love with Jason and unite their powers together! Athena:... Athena: Hold on that is a fantastic idea why didn't I-...oh, actually no wonder I didn't think of that path. I am not an expert in matters of romance and passion.

Like Athena being genuinely saying "the reason I didn't think about it first because I have no interest in romance" hahaha Bonus with Aphrodite as a mother be like:

Aphrodite: Ah! My son! He is so difficult and so uncontrollable! He drives me insane!

[solelii]

Meeting Cerydra at Beyond Time: Eternal Page.

[solelii]

Previous Chrysos Heirs from the Adventures Mission

Verginia (Dux Carminum).

Seneca (Dux Brumalis).

Labienus (Dux Fragoris).

Apollonius (Dux Helkolithist).

[jessecmckeown]

There are ever so many ways to argue

Monge and Apollonius

My grandmother's house was full of books. When I was still relatively small, I found in her basement a book my grandfather (d 1982) must have enjoyed, which I've surely mentioned here before, Triumph der Mathematik by Heinrich Dörrie, translated by David Antin as 100 Great Problems of Elementary Mathematics. It has heaps in it that you wouldn't often encounter anywhere else today, some for reasons you might guess ("Gauss' Altitude Problem", From the time intervals between the moments that three known stars attain the same altitude, determine the moments of the observations, the latitude of the observatory, and the altitude of the three stars. SOLUTION: ...) and some for reasons that I can't guess, but then my brain is wired up something funny.

Monge's Problem

is entry 31. Recall that plane circles are said to intersect normally or perpendicularly if their respective radius segments are perpendicular at the points of intersection.

Monge's problem in the book is then: given three (suitable) circles, describe a fourth that intersects normally with the given three. This happens to be impossible when the three circles intersect too closely, but the geometry is still interesting.

As it happens, the centres of circles normal to two given circles are colinear!

In the figure above, let the circles centred at $A$, $B$, $O$ have radii $a$, $b$, $o$, respectively; and let the right triangles have common leg $y$ and colinear legs $x$ and $z$, respectively. The wealth of right triangles then gives $$ x^2 + y^2 = a^2 + o^2 $$ and $$ z^2 + y^2 = b^2 + o^2, $$ so that $$ (x-z) (x+z) = x^2 - y^2 = a^2 - b^2, $$ the difference $x-y$ depends only on the circle radii $a$ and $b$, and the separation $x+y = |\overline{AB}|$.

The line so defined is sometimes called the "radical line" of the two circles, and if given THREE circles, their three radical lines intersect concurrently -- because if a constructed circle intersect both circles $\alpha$ and $\beta$ normally, and also both $\beta$ and $\gamma$ normally, then it intersects both $\alpha$ and $\gamma$ normally as well. This concurrence also makes it easy to find the radical line in the first place, because as it happens, the radical line of intersecting circles is also concurrent with their intersections. Is that a good exercise? Maybe that's a good exercise.

Tangencies

Two circles are tangent, of course, when they share a tangent line. This means if circles $A$ and $B$, say, are tangent at a point $P$, and third circle intersects $A$ normally at $P$, then it also intersects $B$ normally.

A trivial observation about a circle $X$ tangent to two given circles $A,C$, the centre of $X$ is the same distance from $A$ as from $C$--just as trivially, this means the locus of circle centres tangent to both $A$ and $B$ is probably a hyperbola, unless it's an ellipse, as when $A$ surrounds $B$; HOWEVER! because the common tangent lines of $A$ and $X$, and of $B$ and $X$ are both tangent to $X$, it follows that also their intersection is the same distance from both tangencies -- in other words, that the tangencies are also intersections of $A$ and $B$ with a common normal circle.

We'll combine this with one other useful observation about normally-intersecting circles:

(I'm so terribly sorry that Geogebra makes labels so awkward. I could tug them around, but there are limits to its utility) Drawing any line ($h$, in yellow and red and blue) through one of the intersection points ($B$) defines the bases of two isosceles triangles ($ABD$ and $CBE$); because the circles intersect normally, the base angles of the red and blue triangles are complementary; and because the base angles at $D$ and $E$ are respectively congruent, they are still complementary and so are the adjacent angles of a right triangle. Let's draw some Isosceles triangles into the previous picture.

Because of that right triangle, we can describe the relation of the two tangency points quite neatly, in terms of the line $ST$ joining the centres of $A$ and $C$: they subtend chords from $S$ and $T$, respectively that extend to intersect on the radical line.

These Correspondences be Projective

Given THREE circles, as already discussed, there are three, concurrent radical lines, and so one can chase the tangency points, as in the last figure, around the three circles; usually, we won't end up in the same place we started...

So, starting at point $T$ on circle $a$ (who picks these names?? Why aren't the choices consistent from one figure to the next????) in brown lines we relate it to a compatible point on circle $b$, and in yellow lines relate that to a compatible point on circle $c$, and finally with the green lines relate it to $T'$, again on circle $a$. Each monochrome map is a construction called a perspective map of "ray pencils", and the colour-changing maps are congruences of ray pencils (what's a "ray pencil"? Dörrie is vague on this point -- he is writing as a historian as much as anything, and his sources predate the Everything Is A Set heresy. For us, it's a set of lines through one point). Because the individual maps are all projective(geometry) (They preserve cross-ratios, nothing to do with lifting diagrams) their composite (oh! it's a category after all!) is again projective and as it happens, Dörrie also includes an entry on finding the FixedPoints of such a map! That's a construction by Steiner. It's not bad.

I've also drawn in red the circle through the first three points of this construction, but you can probably tell that it does a little bit more; and figuring why is probably a good Exercise; also, figuring that, since it does do that bit more, the particular projectivity constructed is even an involution. (There's something else in the same part of Dörrie's book, called "Desargues' involution theorem", but what HE means by "involution"... I'll have to read it again.)

So we put these things together and, well, here's a sample construction of an Appolonius tritangent circle:

the black circles are given -- we've suppressed the construction of their three radical lines, but other than that all needed lines and their useful intersections are shown. The bold maroon circle is one of the eight that are tangent to all three given circles. The blue circle is needed to find the fixedpoints of the composite involution on the circle $c$.

Commentary

If we're being strict Euclideans (Socratic Euclideans... cf. his remarks on the solved Delian Problem) then actually, the construction shown above has several shortcuts in it --- it is missing several construction arcs and lines. The geogebra "midpoint" primitive is used a good few times, and so is "circle through three points". Perhaps you notice, about this construction (or any lengthy construction...) It's more than a bit messy. The lines cluster together in awkward ways. They're often nearly parallel. You can imagine that, if you tried to do this by hand, it'd be very difficult even keeping track of Which Line Was Which, and drawing them with sufficient precision would be a Chore.

Now, this isn't the solution that Dörrie chose to showcase in his book --- that was Gergonne's solution, which is a good one, too. I've assembled this method because Gergonne's Approach also uses Monge's Radical Lines (so they're going to be on your page anyway) and to show how much more they do than just locate Normal Circles.

Here is Gergonne's solution, allowing ourselves roughly equivalent shortcutting. I'll let you decide if it's tidier. Here is how Dörrie summarizes Gergonne's construction:

The line is the "Similarity axis" of the given circles, the subject of d'Alembert+Monge's theorem, and it's not an accident that it is also the Radical Axis of the two constructed circles.

Perhaps I might blether on a few more posts about this stuff.

[sapphicstupidity]

reading ptolemy, copernicus, and kepler in math this year made me realize just how much I miss Euclid. I've always been interested in astronomy but it was just a hard switch, after Euclid finally made math make sense to me, I found it hard to engage with ideas outside of his logical organized universe--with astronomy everything just seemed so messy, and I couldn't really understand why.

i was a little worried about moving on to Apollonius because my math tutor stressed how difficult people find it, and even though it goes back to Euclid's format and logical progression of ideas, Apollonius isn't nearly as straightforward as Euclid. But we just started Conics and maybe it'll get harder going forward but honestly it's good to be back, everything makes sense again.

[aestheticallypleasinganimeboys]

Apollonius - Fairy Fencer F