Cat eyes, unit circle, hybrids, composites, and a relational model of poi

So I've posted this on a few other forums and have a simulation up on youtube. I figure it would be good to post here as well.

Unit Circle:
I've started to think of the circle your hand (and indeed the poi head) traces when isolating as the poi 'unit circle'. It's diameter is equal to your poi length. At a given poi length, it is the smallest circle both your hand and the poi head can make. If your hand circle gets smaller the poi head's gets bigger and vice versa. If a 1 unit= the radius of the circle, then the unit circle diameter is 2 units... so your poi is 2 units long.

Cateye:
I've been playing with 'Cateyes' lately. A 'Cateye' is a 1-beat antispin. It is not, however, the linear isolation analogue to a staff antispin cross, which is another 1-beat anti spin.

The Cateye is defined by your hand moving in a unit circle (2 units). The head is spinning in the opposite direction and traces an ellipse who's major axis is 6 units (3x poi length) and who's minor axis is 2 units (1 poi length). The center of the poi body moves in a linear motion 4 units long (2x poi length).

Hybrids:
Classical hybrids arose because you can substitute an extension for an isolation and vice versa.

Once people started to do butterfly hybrids, they ran into problems, having to do polyrhythmic antispun patterns to keep the relationship of hands in phase around the hand circle or 180-deg out of phase. Thus you end up with the more open ended definition of hybrid: using "2 different driving styles". Personally I like to refer to the overarching approach of 'different driving styles' as composite spinning.

You may notice that a Cateye, isolation, and an extension all share the same center of rotation for the hand circle: They are all using unit circle for the hand. This means that you can substitute a Cateye into hybrids as well. This means that we can now explore all the combinations of hand and poi spinning directions in hybrids, as 1-beats.

Here is a simulation I've posted on youtube:
http://www.youtube.com/watch?v=aWw8oe5DACg

In the video I reference Olive's notation for the possible combinations of hand and poi directions and timings. The "Alien Code" is short hand for the configuration of all the relationships going on.

Relational modeling of poi:
Presented is the relational model I'm using. It doesn't yet deal with quarter-time or polyrhythm, as I am trying to start simple and expand from there. Later I'd like to incorporate it with a relational model for crossover points, plane facings, and zones around the human body (maybe frames, but I have a hard time understanding Rich'eese and haven't spent the time to really get it yet).

When relating 2 poi there are 2 underlying attributes, each with 2 possible basic values: [Direction of Spin(0: same or 1: opposite), phase(0: same-time or 1: split-time)]

This can be applied to the relationships of:
a)1st hand vs 1st poi
b)2nd hand vs 2nd poi
c)1st poi vs 2nd poi
d)1st hand vs 2nd hand
e)1st hand vs 2nd poi
f)2nd hand vs 1st poi

Now if you have the same values in a-f, you have a high level of symmetry in poi "configuration space". This gives you 4 different non-hybrid movements:
same-time extensions
same-time isolations
split-time extensions
split-time isolations

The 4 "same poi vs same hand" relationships determine the unit circle "driving styles":
[0,0]=extension
[0,1]=isolation
[1,0]=vertical cat eye
[1,1]=horizontal cat eye

We are all familiar with the 4 poi vs poi and hand vs hand spin but lets write it out anyway:
same direction, same time [0,0]
same direction, split time [0,1]
opposite direction, same time [1,0]
opposite direction, split time [1,1]

In Olives model he applied the 4 to hand/arm motion, but he only applied direction and not timing to poi motion, assuming that they naturally fit in certain ways (ie polyrhythms). In the case of some of the hybrids with more uniform/symmetrical configurations, if you change the poi timing they turn into one of the 4 non-hybrids from above. However, the more complex ones, that are more asymmetrical in configuration space, have different configurations based on timing: ie a vertical cat eye as opposed to a horizontal cat eye.

I'm still digesting and grokking the importance of poi vs opposite hand, as anything more than an epiphenomena, but it feels like it is another angle that is pertinent.

Composite spinning:
I like to think of hybrids in the more rigorous way outlined above. In other words as 'Composites' that work within the 'Unit Circle'. To me 'Composite Spinning' is the overarching relational exploration of poi, where you have higher degrees of asymmetry, which encompasses the idea of spinning with "different driving styles".

ending thoughts:
I haven't yet rendered sims of horizontal Cateyes, or Cateye vs isolations yet. They are pretty well hard, and I'm still learning to cleanly spin the cateye hybrids that I show in the video. Of course other silliness should start to arise, like Cateye hybrid butterfly weaves and such.

Let me know what you guys think,
-Alien jon