From Wed Mar  7 12:04:11 GMT 2001
Article: 83010 of rec.juggling
From: "Denis Paumier" <>
Newsgroups: rec.juggling
Subject: body tricks theory
Date: Wed, 7 Mar 2001 12:59:01 +0100
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Hi rec.juggling,

This is an essay about modelling trajectories of objects around the human
I was inspired by topology and knots theory, especially "invariants" for
 Advices and comments are welcome, it's easier for me by e-mail.

We begin the story with a juggler character after a cartoon crash (the back
stuck on a wall, his body being completely flat, so we can see the his eyes
and his palms); we have the shape of a star with five branches (two legs,
two arms and the head). Let's continue those directions by an infinite line,
so we have five distincts regions marked on the wall. In the wall we can dig
a hole in each of the regions.
 Now let's simplify the situation: mathematically we can reduce it to a plan
divided in five regions each being punched by a hole; we also forget gravity
(balls can now move just by the will of the juggler).
 The space is divided by the plan in two regions: forth and back. We allow
the balls only to be in contact with the palms of hands, so the balls can
only take off to the forth region and land from the same, and we also allow
it to change region (front or back) only by passing through a hole.
 A ball can then take off and land while staying only in the front region,
or it can go through a hole to the back region and come back to the front
using a second hole before landing (it can be the same or a different one).
It can even pass through a hole four times, or six,...
 Let's give a number to each hole: 0 for the one between both legs, and then
clockwise when we watch the juggler (anticlockwise for the juggler itself):
1 for the hole between right leg and right arm, 2 for the hole between right
arm and head, 3 for the one between head and left arm and 4 between left arm
and leg. Then we can write down every combination with two holes:
 00 01 02 03 04
 10 11 12 13 14
 20 21 22 23 24
 30 31 32 33 34
 40 41 42 43 44
 Call back the gravity and the human body. Of course you've got to move to
make the ball do the trajects. But I suppose you return to the basic
position between every traject, even if you don't really do it. I suggest to
practise every traject throwing with right hand back to right, from right to
left, from left back to left and from left to right. Some of the trajects
are common, some you wouldn't have thought of and for some others you need
to think about how to "translate" because of constraints of the body.
 Here are some examples (R is right hand and L is left hand):
 R10L: under the leg
 R12L: shoulderthrow
 R13L: backcross
 R11R: shoulderthrow , catch in the same position
 R00L: just bend and juggle between your legs
 R40L: find it by yourself
 You can find variations for each traject, based on the fact that a traject
has three parts (except the "null" traject which stays in the front zone):
two of them are taking place in the front zone (the beginning B and the
ending E) and the third is contained in the rear zone (the middle part M).
Then you throw in part B and catch in B, M or E; throw in M and catch in M
or E; throw and catch in E. By "throw" I mean that the ball can have an
aerial moment or not (you can sometimes just give it to another).
 Notice that hole 1 is the ymmetric of hole 4 and 2 the symmetric of 3 (and
0 is symetric to itself), so to find the symmetric of a traject you only
need to replace each hole by its symmetric and its extremities by their
symetric (right hand or left hand). You can also find the reverse traject
just reading it backwards.
 Let's play with extentions of these principles: you can try more
complicated trajects and different parts of the body.
 Since then we limited ourselves to use only the plams of hands. But any
other spot on the body is worth using, you just have to know if it's on the
front side of the body or the rear side. If a ball goes from one side to
another, the traject will pass through an odd number of holes (mainly one or
three in real world), if not the traject will have an even length. You can
also use one of those three manners: throw, give or roll a ball on the body
to the next spot.
 Example for a club going from right hand to right foot and passing under
the leg: RH 10 RF
 For a ball going from right hand to "the cradle" on left: RH 3 LC or RH 143
 Let's study trajects with four holes. Since there are many theoritical
possibilities and many of them are not really humanly possible (we'll talk
about simulators later), I'm just going to talk about certain families. You
have the one of double trajects, built by sticking a traject with itself:
R1414L, R1010L,... The family of trajects in which the first hole is the
same as the last one: R1431R, R0430R,...The one you can tell by words such
as "throw under X+catch under Y": "throw under the leg+catch behind the
back" is R0424, "throw under the arm+catch under the leg" is R4340R,...
 You can even try six hole trajects: L012340R. Making a circle with an arm
(leg?) around a ball before catching it may add 21 (or symmetric or reverse)
to a traject. Making a pirouette could add 23...
 Now we shall make a call to vanilla siteswap. The numbers tell us about
when a ball is going to be thrown again, and because we consider only two
hands constantly alternating, we can see in which hand the ball is going to
land. If we precise the first throwing hand of a sequence, then we know from
which hand any ball is thrown and to which hand it goes. So we don't need to
precise it on the traject any more.
 I tried to find some less confusing way to speak about trajects, especially
while talking about siteswaps, numbers spring in every direction. So I tried
to give a name to each hole: the one between BOth Legs is called BOL, the
ones between Arms and Legs are called AL and the ones between Arms and head
(I tought Crane was more convenient for its initial) is called AC. You can
preceed by OP, which means OPposite relating to the throwing hand or to the
previous hole. Example: under the legs is ALBOL, under the arm is OPALAC,
behind the back is ALOPAC,....
 This way we can add extensions to siteswap. I tried to write down for
example the family of "under the arm" tricks, using alac, acal, opalac,
opacal (I'll write that in small letters since I hope it's not confusing; I
don't think there are many other words that sound like that in english)
which are the opposites and reverses of 12, using different hand
combinations. We'll also use "in " and "out" for inside or outside throws
with this definition: throwing inside is throwing from one hand being on its
normal side to the opposite side of the next ball to be caught or throwing
from one hand being on the opposite side to the same side of the next ball;
throwing outside is  throwing from one hand being on its normal side to the
same side of the next ball or throwing with one hand being on the opposite
side to the opposite side of the next ball. Sorry for the painful sentence.
 Let's play with extending the standard cascade in HTML-style: "chops"
3<opalac>, "catch under the arms" 3<acal>, "shoulderthrows" 3<alac>,
"penguin catches" 3<opacal>. Then pattern with a period of two beats: kind
of shower 3<opalac>3<out>, Laurent Pareti's 3<opalac>3<alac>, juggling with
hands crossed 3<acal>3<opacal>,...Patterns of length three, which lead us to
messy patterns: 3<acal>33; 3<acal>3<opacal>3; 3<albol>3<bolopal>3;
boston mess: 3<out acal>3<out>3<in acal>3<out opalac>3<out>3<in>;
mills mess: 3<out acal>3<opalac>3;
half mess: 3<out acal>3<opalac>33...
You can also try: 4<acal>41; 441<alopal>; 441<alopac>; 53<acal>1;...In
general case when you have a 1 in a pattern you can do it  behind the back
or under the leg, and when you have a 31 you can catch the 3 under the arm.
In the great finnish Peapot Videos I saw a surprising 1<opacal> and
 To the simulator makers: would it be possible to simulate bobytricks with
those ideas? I imagine possible tricks with more or less gravity, randomly
selected trajects on randomly selected siteswap patterns and virtually
extended body flexibility…Let me know about it! Have fun.

Denis Paumier
Check Les Objets Volants' page