
The hairy ball theorem - acangiano
http://en.wikipedia.org/wiki/Hairy_ball_theorem
======
samlittlewood
Thus the popularity of unit quaternions in 3D simulation and graphics. They
cover the sphere twice, so each rotation can be represented by two possible
values.

My hack minds eye view of this is that you take two combed layers with nulls
at the poles. The rotation then jumps between layers as it crosses the poles.

~~~
zkz
Could you explain in more detail? Why do people in 3d simulations and graphics
care about the existence of a point where the function value is 0?

~~~
samlittlewood
They want a rotation representation that:

1) can be accumulated (quaternion multiplication) and interpolated (slerp)
'smoothly',

2) and in a way that does not depend on the absolute orientation.

\- Smooth:

If you dive in and start using 3 angles (yaw,pitch,roll) to represent 3d
rotation, then the rotations are like the hairy ball - however you order them,
there will be values where you can't easily step to another physically very
close rotation without a huge change in values - the discontinuities in the
hairy ball.

This is classically called 'gimbal lock', where a gimbal mount is a physical
manifestation of angle representation. In simulations, it might show itself as
strange behaviour when pitched up/down close to 90deg, or an interpolation
between apparently close rotations going wild.

By using unit quaternions (4 values), and avoiding the discontinuties (but now
having two representations for each physical rotation) it becomes easy to
acuumulate the rotations that come from a flight model, animation, user etc.
without out ever getting stuck in a corner.

\- Independant of absolute rotation:

A related point is that an incremental rotation by multiplying quaternions
will always produce the same relative rotation irrespective of absolute
attitude. You can see angle based represetnations going wrong when, for
example, a camera control system offers buttons that are suposed to be
left/right/up/down relative to the camera, but produce varying results as the
camera rotates.

It's possible to do the same things with 3x3 matrices - but it is a bit more
tricky to keep them normalised over time, and obviously takes more storage. A
typical solution to all the above angle problems in a simulation would be to
convert (yaw, pitch, roll) to a matrix, accumulate, then convert back. My
brief experience of aerodynamics texts is that all the equations are
obfuscated with unpicking and rebuilding angles. It's a bit like doing
arithmentic with Roman numerals - possible, but painful.

~~~
zkz
Thanks for taking the time to explain this!

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Retric
I just added this to the comment about the weather:

"This is not strictly true as the air above the earth has multiple layers, but
for each layer their must be a point with zero horizontal windspeed."

Feel free to edit it.

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graphene
I think it would be better to delete the eintire paragraph above your
addition, as the notion that this theorem guarantees at least one cyclone
somewhere on earth is ridiculous.

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byrneseyeview
Is it wrong because the earth is not a perfect sphere, or for some other
reason?

~~~
ars
It's wrong because this theorem only applies to continuous vectors. However
wind can simply fade (i.e. become slower), and is thus non-continuous.

Edit: I'm getting beaten up in the downmods, but no one has shown me that wind
is continuous.

~~~
RiderOfGiraffes
It doesn't apply to continuous vectors, it applies to continuous vector
fields. There is a difference, and it does matter.

And I would be interested to see how you can construct a non-continuous vector
field from an incompressible (which air at these speeds effectively is) fluid.
Your graphic and description do not make sense - they do not allow that,
macroscopically, air is a fluid. If it travels, it has to go somewhere. It
can't simply stop at a boundary, it has to change direction, and such changes
of direction cannot be instantaneous. This is why in electronics we need to
deal with signal reflection, over-voltages, and similar phenomena.

Additionally, continuous technically does not mean "no large jumps", although
that's how the technical definition was inspired, and how most people
visualise it. In particular, it's possible to create a function that's
continuous at every irrational, and discontinuous at every rational.

It's certainly true that the theorem is dealing with a theoretical
approximation to a messy, physical situation, but broadly speaking it's
applicable. It says that at the Earth's surface there is always at least one
place where the horizontal component of the air movement (wind velocity) is
zero. Errors are often made when trying to make folksey explanations, and it's
the interpretations that often have errors. The theorem is true, applicable
and in some cases, useful.

~~~
ars
I thank you for an explanation and not just more downmods.

To explain the wind graph - imagine very low air pressure at the points where
it stops. You wrote incompressible, but that's not actually the case for wind
(although yes the wind doesn't cause compression, but rather the reverse).

"one place where the horizontal component of the air movement (wind velocity)
is zero"

If that's what it says, then yes, I agree that is true for wind. It didn't
seem to be what it was saying though, but I guess I misunderstood it.

It said cyclone, i.e. wind moving in a circle, and that is just not correct.
Wind can simply move out radially in all directions in straight lines from
that point, without making a circle.

That point of course is where the velocity is 0.

To quote "(Like the swirled hairs on the tennis ball, the wind will spiral
around this zero-wind point - under our assumptions it cannot flow into or out
of the point.)". This is not true. Air can flow out of a zero point - just
heat it up, and air will flow out of it.

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RiderOfGiraffes
> This is not true. Air can flow out of a zero point - just heat it up, and
> air will flow out of it.

I'm having a _lot_ of trouble visualising this. There is no air at a point. A
point, by definition, has no volume. Are you manufacturing air?

I think you are using some definitions that are completely at odds with what
everyone else, including me, are using. No doubt if we stood in front of a
whiteboard you could make yourself clear quickly, but almost everything you
have said is, according to my model of how the world works, wrong.

I'd like to understand you, but I suspect that's never going to happen.

I know the theorem - I proved a generalised version of it as a base case for a
much bigger result. I haven't bothered to read most of the comments because
usually the whole thing is mis-quoted or mis-interpreted, but I just had to
say something to try to understand you.

It's true that the theorem does not require a cyclone. The theorem can imply a
zero point with the vector field radiating from it, but in a conservative 2D
fluid flow that can't happen. In a 3D flow you can get that effect on the
surface as the fluid descends to that point and then spreads, but that's
different.

Perhaps you're responding to incorrect "interpretations", perhaps you're right
and I just don't understand you, but you're really not making yourself clear.
Either that, or you're wrong.

~~~
ars
"There is no air at a point. A point, by definition, has no volume. Are you
manufacturing air?"

Sorry. Assume an area, heat it up and wind flows out of it. However the air in
the area itself is moving, so conceptually all the air is moving out of the
point in the center of it. (It's not really, it's moving out of the area, but
all the vectors point away from the point, so that's what it looks like.)

"It's true that the theorem does not require a cyclone."

Thanks. That's really all I was arguing about.

The thing with continuous and cyclone: I was assuming, that people were
saying, that the wind _always_ has to move - even if in a circle. And I was
saying, no, it doesn't have to move, you can have a still area, and wind
radiating out of it (or into it).

If I am correct about that, then please edit the wikipedia article to remove
mention of cyclones.

Why do you say that can't happen in a 2d fluid flow? Why does it have to be a
cyclone? My understanding of weather is you have a large area, you heat it up,
and wind flows out of it - but there is no cyclone. (I guess with fluid flow
you are assuming there is no way to manufacture fluid, but with wind you can
since heat will "create" more of it.)

Tell me if I'm wrong here:

The hairy ball theorem assumes there is hair everywhere, so you have to have a
cyclone at the poles. But with wind there are spots without hair, so the
theorem just doesn't apply to wind.

~~~
cma
An idealized 2d fluid flow is non-compressible, so heating it up at a point
does nothing.

~~~
ars
But air is compressible. And I am talking about air.

(Unless this post was answering my question, in which case thanks.)

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Dilpil
Any other interesting applications of topology to physical phenomenon? I can
only seem to find applications to other areas of math.

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beaker
The theorem lacks a certain aesthetic beauty. Apply Occam's Razor to create a
smoother, more palatable package.

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jackdawjack
neat, this is very important in classical mechanics. Orbits of your system
trace out tori in the phase space, the cool thing is how the tori break up as
you perturb your system from integrable to non-integrable. Chaos, without
dissipation.

~~~
davi
more/links?

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IsaacSchlueter
Hahahah...

Hairy balls.

