strathspey Archive: Cosmic Reel 2

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Cosmic Reel 2

Message 21022 · Thomas J. Senior · 12 Apr 2000 18:13:56 · Top

Hello again,
I've checked out the reference for Montgomery and Chenciener's solution
to the 3 body problem, and have, with the help of an able physics student,
set it up on a gravity simulation program (Gravitation 5.0, shareware for
Mac). The animation is neat to see, but the speed is not constant, which
might upset some who would like to see the reel of 3 danced with perfectly
constant speed. The planets move fastest when moving through the middle,
and noticable slower on the farthest extreme before being pulled back
toward the center.
If you like the parameters for Gravitation 5.0, i'll be happy to email
them to you privately.

Dance happy,
Tom

Those who can, do. Those who understand, teach.

Thomas J. Senior
New Trier High School 1232 St. Johns Ave
385 Winnetka Ave Highland Park, IL 60035-3425
Winnetka, IL 60093-4295 847-433-8704
847-784-6739 seniort@nttc.org
Science FAX 847-501-6408

Cosmic Reel 2

Message 21025 · Mettler, Klaus · 13 Apr 2000 09:17:20 · Top

Hello,

I've also a solution to the 3 body problem on our homepage,
sub-site "Termine". In this animation the speed of the planets is also
not constant, but the planets move only faster in one direction through
the middle (every dancer can see in which).
If you have a look on my real of three - animation, you can change the
speed between 1 and 10 and the shape of the reals with the buttons R1,=20
R2, R3 and R4. If your computer isn't fast enough you could get =
collisions
in the center on speed 5 or 10.

Happy Dancing

Klaus Mettler
Stuttgart / Germany
Email: mailto:klaus.mettler@gmx.de=20
Homepage: http://www.tamburin.de/
Scottish Country Dancing in Stuttgart, Schw=E4bisch Gm=FCnd, Denkendorf =

Cosmic Reel 2

Message 21029 · ron.mackey · 14 Apr 2000 00:06:37 · Top

. The animation is neat to see, but the speed is not constant, which
> might upset some who would like to see the reel of 3 danced with perfectly
> constant speed. The planets move fastest when moving through the middle,
> and noticable slower on the farthest extreme before being pulled back
> toward the center.

Hi, Thomas
That sounds most reasonable and in line with my experience!!
Cheers, Ron :)

< 0 Ron Mackey,(Purveyor of Pat's Party Pieces)
'O> Mottingham,
/#\ London. UK.
l>
Ron.Mackey@btinternet.com

Cosmic Reel 2

Message 21039 · James Smith · 14 Apr 2000 21:26:38 · Top

"Mettler, Klaus (LFUST)" wrote:
>
> Hello,
>
> I've also a solution to the 3 body problem on our homepage,
> sub-site "Termine". In this animation the speed of the planets is also
> not constant, but the planets move only faster in one direction through
> the middle (every dancer can see in which).
> If you have a look on my real of three - animation, you can change the
> speed between 1 and 10 and the shape of the reals with the buttons R1,
> R2, R3 and R4. If your computer isn't fast enough you could get collisions
> in the center on speed 5 or 10.
>
> Happy Dancing
>
> Klaus Mettler
> Stuttgart / Germany
> Email: mailto:klaus.mettler@gmx.de
> Homepage: http://www.tamburin.de/
> Scottish Country Dancing in Stuttgart, Schwäbisch Gmünd, Denkendorf

For an animated reel of 3 at constant speed check the link below then
select Reel of 3.
http://www.members.home.net/dancimation/scd/html

James Smith, Vancouver

Cosmic Reel 2

Message 21072 · Alan Paterson · 19 Apr 2000 13:58:28 · Top

James Smith wrote:

> For an animated reel of 3 at constant speed check the link below then
> select Reel of 3.
> http://www.members.home.net/dancimation/scd/html

James,

Sorry, but when I try to access this page, I get the 'Page not Found' error.

Alan

--
Alan Paterson
Berne, Switzerland
mailto:alan@scottap.com

Cosmic Reel 2

Message 21073 · DonaldMackay · 19 Apr 2000 14:35:19 · Top

Alan

It should have been

http://www.members.home.net/dancimation/scd.html
NOT
http://www.members.home.net/dancimation/scd/html

and the animations look great!

Donald Mackay

Email: DonaldMackay@uk.ibm.com
Please respond to strathspey@tm.informatik.uni-frankfurt.de

To: strathspey@tm.informatik.uni-frankfurt.de
cc: (bcc: Donald Mackay/UK/IBM)
Subject: Re: AW: Cosmic Reel 2

James Smith wrote:

> For an animated reel of 3 at constant speed check the link below then
> select Reel of 3.
> http://www.members.home.net/dancimation/scd/html

James,

Sorry, but when I try to access this page, I get the 'Page not Found'
error.

Alan

--
Alan Paterson
Berne, Switzerland
mailto:alan@scottap.com

Cosmic Reel

Message 21171 · Oberdan Otto · 7 May 2000 01:32:38 · Top

>[Tom:]
> I've checked out the reference for Montgomery and Chenciener's solution
>to the 3 body problem, and have, with the help of an able physics student,
>set it up on a gravity simulation program (Gravitation 5.0, shareware for
>Mac). The animation is neat to see, but the speed is not constant, which
>might upset some who would like to see the reel of 3 danced with perfectly
>constant speed. The planets move fastest when moving through the middle,
>and noticable slower on the farthest extreme before being pulled back
>toward the center.

I bet you thought this discussion was finished!

I have been watching people dance reels of three quite a bit lately.
Sure enough, the reels are tilted with respect to the set lines. So I
have been asking myself "how much is the tilt?". It looks like it is
somewhere in the range of 15 to 25 degrees, but what is it really
(even though most dancers don't really care, inquiring minds want to
know)? Since I cannot easily go onto the floor and mark the tracks of
the dancers in an actual reel, I must use a model.

[For those people who don't want to read the rest of this, but want
to know THE ANSWER, here is the spoiler: If the 8 is 3 times as long
as it is wide (typical??) then the tilt angle is 22.7 degrees.]

As nifty as the solution to the gravitational three-body problem is,
it does not describe what dancers do. In the gravitational problem,
the bodies speed up when they get close to each other. I think
dancers, who are self-propelled and not governed by mutual
gravitational attraction, tend to do the opposite, being cautious to
avoid collisions!

So how do I model the reel of 3? The general case of reels of 3 with
dancers going at varying speeds and crossing through the center at
varying intervals is just too complicated for me. So I will make some
simplifying assumptions that I think are pretty close to what dancers
actually do. First I assume that all three dancers dance at the same
constant speed around the entire reel and that all dancers dance on
the same figure of 8 track. Next I assume that the dancers cross
through the middle at equal intervals in time. What this means is
that each dancer is 1/3 of the total path length of the reel behind
the dancer s/he is following.

So far, so good. If at the start of the reel, one dancer is at the
crossover point, then one dancer is 1/6 of the total path length from
reaching the crossover point and one dancer is 1/6 of the total path
length beyond the crossover point. Since each loop of the 8 is 1/2 of
the total path length, the dancers are NOT at the extremes of the 8
(which are 1/4 the total path length from the crossover point) at the
beginning of the reel! Although the three dancers are in a straight
line (the set line or the line directly across the set), that line is
not the axis of the 8. The axis of the 8 is tilted from the set line.

OK, Oberdan, that's what you told us at the beginning. So what is the
tilt already? Patience, please. First we have to describe the shape
of the 8. I think the dancers actually trace a track on which the
curvature changes continuously. For the mathematicians among us, all
the derivatives of the path curvature are continuous. Well, sorry
folks, but that is also too complicated for me. However, I think I
can make a pretty good approximation by using just circular arcs (for
the ends of the 8) and straight line segments for the crossover part
of the 8. This means constant, non-zero curature along the arcs and
zero curvature along the line segments--not very continuous. I will
make one stipulation for a smooth path without kinks: the ends of the
circular arcs that join to the straight line segments are pointing in
the same direction as (tangent to) the straight line segments. With
that stipulation, the 8 looks nice and smooth, and it is uniquely
specified by the ratio of its end-to-end length to its width. With
this description of the 8, the smallest Length/Width (L/W) ratio is
2. There is no upper limit.

Although not rocket science, the mathematical analysis is a bit messy
for this forum. However, here is an approximate formula that is
within 1 degree of the "correct" value for L/W 2.5 to 5.9, and within
0.08 degree of the "correct" value for L/W from 3.2 to 5.1:

Tilt Angle = 34.6 - 4.03 x (L/W) degrees

And here is a table that shows that the formula is pretty good for
any practical L/W ratio:

L/W Correct Tilt Formula Tilt

2.0 30.00 26.54
2.5 25.34 24.53
3.0 22.67 22.51
3.5 20.50 20.50
4.0 18.49 18.48
4.5 16.48 16.47
5.0 14.48 14.45
5.5 12.84 12.44
6.0 11.54 10.42

The case of L/W=2.0 is actually not practical: it has the dancers
crossing through the middle of the 8 perpendicular to the axis of the
8. Since the 8 is tilted at 30 degrees, this means that the dancers
would actually be backtracking or retrograde when crossing through
the middle for one direction of crossing. I have never seen dancers
do that even for very fat reels. I think the range of 2.5 (very fat,
tilt=25.3 degrees) to 4.0 (very elongated, tilt=18.5 degrees) covers
the practical range pretty well.

My conclusion is that over the entire range of practical reel shapes,
the tilt angle is significant (within a few degrees of 21 degrees).

Bear in mind that dancers are very flexible creatures, so if you drew
a non-tilted 8 on the floor, they could dance a reel of 3 on that
track, but they would have to slow down and speed up a lot to avoid
collisions. Left to their own devices, dancers would prefer to dance
at a constant speed and leave as much space as possible for the
crossovers, so the 8s will tilt!

Cheers, Oberdan.

184 Estaban Drive, Camarillo, CA 93010-1611 USA
Voice: (805) 389-0063, FAX: (805) 484-2775, email: ootto@ootto.com

Cosmic Reel

Message 21172 · Dianna Shipman · 7 May 2000 02:17:03 · Top

Does it matter if the reels are done in 6 bars or 8 bars of music (not being
a mathematician)? or the speed of the music? I've noticed with slower
music (even slower jig and reels) the reels become fatter and with faster
music thinner (since you're trying to cover less ground in the time
alloted).
Dianna

Dianna L. Shipman
diannashipman@worldnet.att.net
PMB 134, 1436 W. Gray
Houston, TX 77019-4946
Scottish Country Dancing and More
web page: http://home.att.net/~diannashipman
phone: 713-522-1212
----- Original Message -----
From: "Oberdan Otto" <ootto@ootto.com>
To: <strathspey@tm.informatik.uni-frankfurt.de>
Sent: Saturday, May 06, 2000 4:30 PM
Subject: Re: Cosmic Reel

(snip)
> Bear in mind that dancers are very flexible creatures, so if you drew
> a non-tilted 8 on the floor, they could dance a reel of 3 on that
> track, but they would have to slow down and speed up a lot to avoid
> collisions. Left to their own devices, dancers would prefer to dance
> at a constant speed and leave as much space as possible for the
> crossovers, so the 8s will tilt!
>
> Cheers, Oberdan.
>
> 184 Estaban Drive, Camarillo, CA 93010-1611 USA
> Voice: (805) 389-0063, FAX: (805) 484-2775, email: ootto@ootto.com
>

Cosmic Reel

Message 21173 · Oberdan Otto · 7 May 2000 04:33:03 · Top

>[Oberdan:]
> > Bear in mind that dancers are very flexible creatures, so if you drew
> > a non-tilted 8 on the floor, they could dance a reel of 3 on that
> > track, but they would have to slow down and speed up a lot to avoid
> > collisions. Left to their own devices, dancers would prefer to dance
> > at a constant speed and leave as much space as possible for the
> > crossovers, so the 8s will tilt!
>
>[Dianna:]
>Does it matter if the reels are done in 6 bars or 8 bars of music (not being
>a mathematician)? or the speed of the music? I've noticed with slower
>music (even slower jig and reels) the reels become fatter and with faster
>music thinner (since you're trying to cover less ground in the time
>alloted).

Neither the number of bars nor the speed of the music affects the
reel tilting analysis. BUT when the dancers have less time, they
cover less territory, so the reels become skinner, L/W (length to
width ratio) increases and the reels tilt less. When more time is
available, the reels can become fatter, L/W decreases and the reels
tilt more.

I think this fits your observations.

Cheers, Oberdan.

184 Estaban Drive, Camarillo, CA 93010-1611 USA
Voice: (805) 389-0063, FAX: (805) 484-2775, email: ootto@ootto.com

Cosmic Reel

Message 21180 · Blaine L. Peet · 7 May 2000 17:25:47 · Top

On Sat, 6 May 2000 14:30:52 -0700 Oberdan Otto <ootto@ootto.com> writes:
> [For those people who don't want to read the rest of this, but want
> to know THE ANSWER, here is the spoiler: If the 8 is 3 times as long
> as it is wide (typical??) then the tilt angle is 22.7 degrees.]

OK Oberdan, now for the bonus question:

What is the similarity (or difference) to a reel of three and the
velocity of the 3 balls a juggler uses during a "standard" juggling
routine. At what point are the 3 balls aligned? Please feel free to
make any assumptions you care to!

This inquiring mind is only a wee bit curious. (One of our local dancers
also juggles - it would be a bit amusing to assign him this problem to
solve.)

Blaine Peet
Maryland, USA

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Cosmic Reel

Message 21182 · ron.mackey · 8 May 2000 00:26:25 · Top

> Neither the number of bars nor the speed of the music affects the
> reel tilting analysis. BUT when the dancers have less time, they
> cover less territory, so the reels become skinner, L/W (length to
> width ratio) increases and the reels tilt less. When more time is
> available, the reels can become fatter, L/W decreases and the reels
> tilt more.
>
> I think this fits your observations.
>
> Cheers, Oberdan.

Hi,
I'm not too sure who pointed out the different kinds of covering in
a reel but, to me, it is certain that many reels are distorted
because many are taught to cover with their partner, full stop.
On Friday night at our last class of the season we did Lady Susan
Montgomery from Leaflet 33. Apart from being a very nice, gentle
strathspey with great music, it finishes with six bar reels on the
'wrong' sides 1s giving right shoulder to second corners.
Some of the class have this tendency to cover with their partner so
I emphasized that the covering came from dancing in through second
place and acknowledging the person on the other side and coming
from the other direction with a smile and - dare I say this?- eye
contact. They timed the reels perfectly and it gave a great cachet to
the whole dance.
Cheers, Ron :)

< 0 Ron Mackey,(Purveyor of Pat's Party Pieces)
'O> Mottingham,
/#\ London. UK.
l>
Ron.Mackey@btinternet.com

Cosmic Reel

Message 21183 · ron.mackey · 8 May 2000 00:26:31 · Top

> Neither the number of bars nor the speed of the music affects the
> reel tilting analysis. BUT when the dancers have less time, they
> cover less territory, so the reels become skinner, L/W (length to
> width ratio) increases and the reels tilt less. When more time is
> available, the reels can become fatter, L/W decreases and the reels
> tilt more.
>
> I think this fits your observations.
>
> Cheers, Oberdan.

Hi,
I'm not too sure who pointed out the different kinds of covering in
a reel but, to me, it is certain that many reels are distorted
because many are taught to cover with their partner, full stop.
On Friday night at our last class of the season we did Lady Susan
Montgomery from Leaflet 33. Apart from being a very nice, gentle
strathspey with great music, it finishes with six bar reels on the
'wrong' sides 1s giving right shoulder to second corners.
Some of the class have this tendency to cover with their partner so
I emphasized that the covering came from dancing in through second
place and acknowledging the person on the other side and coming
from the other direction with a smile and - dare I say this?- eye
contact. They timed the reels perfectly and it gave a great cachet to
the whole dance.

I have just read the rest of today's mail on this subject but will
leave this is as an illustration. :)

Cheers, Ron :)

< 0 Ron Mackey,(Purveyor of Pat's Party Pieces)
'O> Mottingham,
/#\ London. UK.
l>
Ron.Mackey@btinternet.com

Cosmic Reel

Message 21238 · Ian Brockbank · 16 May 2000 14:55:13 · Top

Hi Oberdan,

> So how do I model the reel of 3? The general case of reels of 3 with
> dancers going at varying speeds and crossing through the center at
> varying intervals is just too complicated for me. So I will make some
> simplifying assumptions that I think are pretty close to what dancers
> actually do. First I assume that all three dancers dance at the same
> constant speed around the entire reel and that all dancers dance on
> the same figure of 8 track. Next I assume that the dancers cross
> through the middle at equal intervals in time. What this means is
> that each dancer is 1/3 of the total path length of the reel behind
> the dancer s/he is following.

Er, sorry, these simplifying assumptions are, quite frankly, invalid.
You have one person in the middle, and two people trying to swap ends
in half the duration of the reel. The lead dancer has it easy. The
other two ARE TRYING TO BE IN THE SAME PLACE AT THE SAME TIME (in
theory). So what you get is all three people on a lissajou figure
(slower round the end 'cos you can dance as fast when turning), and
the two non-dancers have an extra sinusoidal permutation on their
speed across the centre - one person speeding up then slowing down,
the other doing the opposite - to avoid crashing into each other.

At least, that's how I modelled it for my dance animation program for
my final year degree project...which I must get round to putting up
on my web site as I have been promising various people for years...

Ian
--
Ian Brockbank, Indigo Active Vision Systems, The Edinburgh Technopole,
Bush Loan, Edinburgh EH26 0PJ Tel: 0131-475-7234 Fax: 0131-475-7201
work: ian@indigo-avs.com personal: Ian.Brockbank@bigfoot.com
web: ScottishDance@bigfoot.com http://www.scottishdance.net/
Feed the World http://www.hungersite.com/

Conventional-Covering and Counter-Covering in Cosmic Reels

Message 21174 · Oberdan Otto · 7 May 2000 04:33:18 · Top

Hi,

There is an interesting point about covering pairs of reels of 3 if
the reels are danced as described by my model (constant speed for all
dancers, and evenly spaced crossings through the middle).

The reels (whether parallel or mirror) can be covered across the line
between the two reels if the original middle dancers start moving
toward the same ends of the reels (i.e both toward the top, both
toward the bottom, both toward ladies' side or both toward men's
side). Then the end dancers' counterparts in the other reel will be
crossing through the middle at the same time. Basically, you cover
with the dancer next to you in the other reel. I call this
"conventional covering". If you phrase the reels for evenly spaced
crossing through the middle, you will be naturally
conventional-covered.

If the middle dancers begin moving toward opposite ends of the reels,
it is obvious that the middle dancers cannot cover across the plane
between the reels, but they can "counter-cover" through the center
POINT between the reels. The SAME is true for the end dancers--they
also cannot cover across the line between the reels, but they can
counter-cover with the person at the opposite end of the other reel
through the center point. In these kinds of reels, you covering
counterpart is NOT the person next to you.

Misapplication of the notion of covering can really mess up the reels
and create near collisions. Example: Montgomeries Rant has 8-bar
reels (on the sides) and 6-bar reels (across the dance). BOTH of
these reels are "counter-covering" reels--one should not attempt
conventional covering in them. If you phrase the reels for evenly
spaced crossovers, you will be naturally counter-covered.

Notice that the phrasing paradigm of evenly spaced center-crossings
automatically produces the correct form of covering for both types of
reels.

Cheers, Oberdan.

184 Estaban Drive, Camarillo, CA 93010-1611 USA
Voice: (805) 389-0063, FAX: (805) 484-2775, email: ootto@ootto.com

Conventional-Covering and Counter-Covering in Cosmic Reels

Message 21181 · Blaine L. Peet · 7 May 2000 17:25:53 · Top

On Sat, 6 May 2000 17:29:24 -0700 Oberdan Otto <ootto@ootto.com> writes:

> If the middle dancers begin moving toward opposite ends of the
> reels, it is obvious that the middle dancers cannot cover across the
plane
> between the reels, but they can "counter-cover" through the center
> POINT between the reels. The SAME is true for the end dancers--they
> also cannot cover across the line between the reels, but they can
> counter-cover with the person at the opposite end of the other reel
> through the center point. In these kinds of reels, you covering
> counterpart is NOT the person next to you.

This was precisely the focus of Christine Mair's teachers' workshop at
last month's Capital Weekend here, that she calls "covering on the
diagonal". She had an interesting exercise to demonstrate this, using
"markers" held by each person as they moved through reels of 3 on
opposite sides, so that you focused on your "equivalent" for covering.
Very enlightening.

Blaine Peet
Maryland, USA
________________________________________________________________
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