Dear Jeff and dear All
I was almost certain you will ask me about chuff-chuffs.
As you guess by reading my posts, I can't write english as well as W.S, so I
choose to translate our french "teuf-teuf" by "chuff-chuff". But it was just
an onomatopeia!
Do you remember this celebrated movie: "Once upon a time in the West", a
spaghetti western by Sergio Leone? One of the heros, a gunner named
Cheyenne, always playing harmonica said to one of the villains:
Hey, you, wait a minute.
Let's have a good look at you, Mr Choo-Choo.
It's easy to find you, bastard.
I don't have to kill you now.
You leave a slime behind you like a snail.
Two beautiful shiny rails...
When I was a little boy going to school in commuter trains around Paris just
after the War (remember, you were here!?), I was always fascinated by other
trains on parallel tracks going quickly or slowly and sometimes standing
still for a short time.
I could not guess some sixty years later, I would think about them!
That's why I name this part of triangle geometry:
La théorie des teuf-teufs or chuff-chuff theory.
But it's serious business and it was a very very long time before I became
aware that affine geometry ( and not euclidian geometry) was the proper
framework to study them, to say uniform motions of several points on
different lines in a plane
You can begin by 1, 2 then 3 and so one.
There is a very good book on the theory of uniform motions on a line:
Galilean geometry by Yaglom. In fact the Newtonian relativistic (not
Minkowski relativistic) spacetime for such motions is a plane, the Galilean
plane according to Yaglom.
So I am interested in the galilean spacetime E of dimension 3 for uniform
motions in the plane in which you choose a triangle (eh, eh!) to do
calculus.
I begin with a little example to show usefullness of E.
Given a triangle ABC in the affine plane and 3 uniform motions: t --> a(t),
t --> b(t), t --> c(t) respectively on sides BC, CA, AB, how to find another
uniform motion t --> m(t)on a line L such that this new motion was in
"Crash" (another good movie, is not it?) with the 3 first.
That is, there are 3 values of time t1, t2, t3 such that:
m(t1) = a(t1); m(t2) = a(t2); m(t3) = a(t3).
If you think at a uniform motion t --> m(t) in the plane, its world-line in
E is just a line t --> (t,m(t)).
Two uniform motions of the plane are in crash if and only if their
world-lines in E are meeting!
So given 3 uniform motions, they generate 3 world-lines L1, L2, L3 in E. Now
you look at the quadric of which L1, L2, L3 are in the same set of
generators and the sought motions correspond to the other set of generators.
So there is a lot of geometry and the best is to do calculus with
barycentric coordinates wrt ABC. At the beginning it was very tedious but
not so tedious with the help of Mathematica to compute some 3x3 determinants
or to factor other ugly polynomials.
One of the key man at the start of this theory is V.Thébault who defines the
notion of "équicentre" that is if you look at the 3 motions t --> a(t), t-->
b(t), t --> c(t), you can find in general 3 reals u, v, w with u + v + w =
1 such that m(t) = u.a(t) + v.b(t) + w.c(t) is a fixed point!
I say "in general" for of course there are exceptions!
Notice I have found some other points than the equicentre related to these 3
motions and having some geometric significance, so allowing to do synthetic
geometry and avoiding many tedious calculus.
Of course euclidian geometry is not totally absent and it's a funny game to
see how very famous points of triangle geometry, as Brocard points for
example but they are others,are concerned with the chuff-chuffs. They give
also a new insight to an old euclidian theory, to say the composition of
two direct similarities.
Isotomic and isogonal conjugations are playing a major part in the theory,
that's why in a chuff-chuff picture, I came across this in-conic with
isoconjugation.
Of course, conics and quadrics are also in the core of the theory and that's
why some calculus are so tedious.
Another affine invariant is playing a major part, the signed area. I
underline there is no need of an affine structure to define it!
For example you can look at the signed area S(a(t),b(t),c(t)) of our 3
uniform motions. It is easy to see that S is in general a quadratic function
but there are cases where S is linear and even constant! That's just the
cases where Thebault's equicentre does not exist. If you succeed to draw all
these special motions, you will understand the difficulties and the beauty
of the theory.
I finish with the "big?" theorem of the theory which allows us to do
synthetic geometry after its proof!
I don't know if it is known and maybe some Hyacinthists could tell me about.
Given "in general" four uniform motions t --> m_{l}(t), l = 1,2,3,4, and
gathering them 3 by 3, you obtain 4 (Thebault's) equicentres q_{l}, l=
1,2,3,4.where q_{l} is equicenter of the motions t --> m{i}(t), t -->
m{j}(t),t --> m{k}(t), {i,j,j,l} beeing a permutation of {1,2,3,4}.
Then q_{1},q_{2},q_{3},q_{4} are on a same line and their crossratio is easy
to compute and have a geometric significance.
That's all, folks!
Friendly
François
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