Tue, 05/06/2014 - 11:00pm by Drex **Tags:**

Four months ago, Jon Alvarez asked a seemingly innocuous question on the Poi Chat forum on Facebook that led me to one of the most mammoth undertakings of my adult life: has anybody set down definitions of all the poi moves in one place? The answer is sadly no, but it got me thinking about why that answer was no...beyond whether someone had set up a dictionary or encyclopedia, to the very heart of how we define poi tricks and discuss them online.

There's no doubt that we as a community vastly prefer creating moves to writing down the ideas behind them or even creating falsifiable definitions by which we can judge whether a given trick fits a definition. It tends to be a lot more exciting to come up with the trick than it is to explain what it is. Even when we make attempts to do these things they are frequently hotly contested even among our close friends. In my infinite wisdom I decided to wander into this morass, but it led to a very different place than I'd anticipated. For one, the only way I could think of to formally define many of the poi tricks I knew was to use the most rigid language I knew of: mathematics. Years ago I'd been taught the basics of poi math in a life-changing geek session with physicist Adam Dipert at Wildfire that had ignited my curiosity and I came to realize that using these techniques to describe poi tricks was going to involve educating much of my audience on the the basics of using trig functions to model poi movement.

50 pages and three days later I discovered I'd written down something our community never had before--a reasonably complete guide to using trigonometry to model poi moves as diverse as flowers and toroids, setting down into text knowledge that many of us had learned years ago and took for granted that the rest of the prop spinning world had access to.

I met my goal--defining as many moves as I could through the lense of values for mathematical variables. But what I hope this accomplishes in the macro context of the prop spinning world is encouraging those of us sitting on vast stores of knowledge to write as much of it down as we can and share it with the rest of the world. Doing so ensures the legacy of our discoveries and the ease of the next generation educating itself on our breakthroughs.

In the meantime, I'm publishing this paper both as a Google Document that anyone can access (available here) and to make it easier to search for individual parts of it, publishing it in sections on my blog for the next several days before publishing the whole thing in my own reference section. I hope the reader gets something out of what I've produced here. It was definitely a labor of love. Big thanks to my editors: Jexi, Pierre Baudin, and Adam Dipert for reading through such a vast piece of work and for their excellent suggestions on how to improve it.

The art of poi may be one of the most unique confluences of art, movement, math, and science in the modern era. At its root, the art is based in tribal dances practiced by the Maori of New Zealand for an unknown period before being exported to the Western world to integrate with a variety of martial arts, dance, and other prop manipulation styles. In its current form, spinning poi focuses on the creation of curves in space using a weighted end connected via a flexible tether to the performer’s hand. Despite such simple apparatus, in the past decade poi has exploded in its vocabulary and number of practitioners.

As poi has acquired more practitioners and edged ever closer to a mainstream pursuit, it has also become the focal point of a variety of controversies among its practitioners. One such series of controversies is over the definition of even some of the most basic poi moves. As much of the movement vocabulary for poi has been created on an ad hoc basis or passed down via oral tradition rather than recorded or formalized in deliberate fashion, there exist strong regional differences in definitions, movement vocabularies, and interpretations of how movements interrelate to the point that such conflicts are all but inevitable.

We in the poi world are fortunate, however, in that much of the vocabulary of our art can be modeled using fairly trivial mathematics. Indeed, the vast majority of shapes and tricks we poi spinners produce can be modeled using trig functions that many of us learned in high school. This provides us with two opportunities: the first is that by understanding how the values in these equations generate the moves we are familiar with, we can easily enter alternate values to create patterns we have yet to experience. This expands our vocabulary as well as our understanding of the art. Second: we can find the boundaries of values necessary to produce given families of tricks. It is this second benefit that is most intriguing, for it offers us an opportunity to formalize the definitions of poi tricks in a way that is easily falsifiable and thus finally come to discuss poi within the context of a true shared vocabulary.

Given that this method of modeling poi movement through mathematics is not well known within the poi community at large, I have written this paper with the following two goals:

- Educate the general public and flow community as to how to model poi moves utilizing the trig functions I was taught years ago.
- Create the first formal classification scheme for poi tricks based in mathematical values rather than arbitrary metaphors or imprecise words

In order to restrict the length of what is already a painfully long paper, I am limiting the scope of what I cover only to patterns and tricks that are identifiable when performed with a single poi sans the body of the performer. As such, this paper does not include any information that covers the following topics: hybrids, inversions, timing and direction, etc (I’ve got to have some stuff to save for a follow-up if ever I get crazy enough to attempt a follow-up). With that in mind, this paper will consist of two sections: the first is meant to teach the reader the fundamentals of how trig functions can be used to model poi movement. It will cover trig equations that model a variety of different types of common poi moves. After covering all of these cases, I will use them to generalize an equation from which we can model any and all of the poi patterns covered in this paper (and hopefully pretty much every single poi pattern possible). I will use this equation in the second half of the paper to set down definitions for each category of poi movement modeled in the first half of the paper.

Let’s start with the math.

Poi consists of a small weight on the end of a tether connected to a person’s hand. The usual practice is to perform with 2 poi, one for each hand, though explorations of 3 and 4 poi configurations have become more common in recent years. The materials used for this prop vary widely according to the performer, location, and resources at hand. However, fundamentally all tricks the performer may create with the poi will be based upon the idea of introducing accelerations in various directions through the tether to the head, leading to an abundance of possible patterns.

The dominant approach to poi performance for the better part of a decade has been to spin the poi in patterns that can be derived from vertical curves, presenting the audience with as full a profile of said curves as possible. Given that the most common approaches to these tricks are to induce the poi to travel in a circular or elliptical path around the hand/handle, we can use math designed to describe circles and gradually add layers of complexity as needed to make the equations capable of displaying more complex patterns. First, a few caveats:

This type of math is excellent for describing the simple system of how two points in space relate to each other over time, but not how they interact with other objects introduced into the system. As such, we can use this method to describe how the poi moves in space, but how it moves in relation to the body is in many cases outside the bounds of this type of mathematical modeling. For this system, we will focus exclusively on the patterns the poi head and handle generate over time and not focus as much on how these patterns interact with the body nor how it presents obstacles that specific types of moves are designed to avoid. As such, any and all contact moves, wraps, inside moves, and throws are not covered in this breakdown.

This type of math is excellent for modeling complete poi patterns but is less helpful when modeling partial or composite patterns. While a section exists to describe some elementary versions of these patterns, the complexity of the modeling increases dramatically the more moves we try to understand in sequence.

Modeling poi in this fashion is agnostic of gravity--due to the forces of acceleration placed on the poi head, this frequently is not as important to understanding a pattern as the mathematically perfect version of said pattern is, but it is worth noting as a shortfall of this approach.

This section will work heavily in the fundamental math behind periodic functions. It is not necessary to read this section completely to understand everything that comes after it, especially if the reader already has an operating knowledge of Trigonometry. If the reader would like, they can skip this section to avoid the basic math and move on to how it applies to poi by clicking here.

Trigonometric functions provide the easiest means of modeling the behavior of cyclical functions over time. I am indebted to Adam Dipert for first teaching me how cycloid math could be simplified to provide a good model for poi movement and Will Ruddick for assisting me in creating computer simulations that taught me the depth to which these movements could be modeled using this mathematical approach.

To begin, we need an understanding of sine and cosine functions. To the left is a chart of a basic right triangle defined by angles ABC and sides abh. If we pick angle A as our home base, we can see it has two intersecting sides: b and h. Side h is referred to the hypotenuse as it is the only side that does not intersect with the right angle of the triangle, angle C. The remaining two sides are labeled according to their relationship with angle A: b is referred to as the adjacent side because it, along with the hypotenuse intersects at said angle. Side a is referred to as the opposite side as it is the side opposite angle A.

To define our first trigonometric function, we will establish a relationship between the lengths of sides a (the opposite) and h (the hypotenuse), that is if we divide a by h, we get a relationship referred to as the sine of angle A (sinA=ah). Depending upon the measure of angle A, a and h will have set proportions ranging from 0:1 to 1:1. Sine is usually abbreviated as sin in these equations.

Our second trigonometric function is the cosine function, which has a different relationship to angle A than the sine function does. For cosine, we are dividing b (the adjacent side) by h (the hypotenuse), resulting in the equation cosA=bh. As with the sine function, cosine will have set proportions depending on the size of angle A ranging from 1:1 to 0:1, but the proportion will always be different than sine A except in the special case when a = b, or the two sides are of equivalent length.

Now that we know the basics of these functions, let’s learn a little bit about how we can graph the results of them out and change the resulting graph by adding particular values to different parts of the function.

To start, we will get results for a few values of y=sinx(please note, I am using radians for values of x--to get the same values, ensure that your calculator is in RAD mode, not DEG. To find out why I'm doing this and what radians are, click here):

x |
y 0 0.707* 1 0.707* 0 -0.707* -1 -0.707* 0 |

* For these values I have truncated down a much longer number to only three decimal places

As we can see, even though the value of x continues to rise, the value of y cycles between 1 and -1, passing through numbers in between in predictable intervals. The animated image below displays how this equation is graphed with all values of x between 0 and 2pi:

As you can see, graphing this equation produces a gentle, wave-like curve that is known as a sine wave. The sine wave goes no further than 1 unit positive or negative away from the x-axis and begins to repeat its results after 2. We are going to introduce two terms here that will be used as placeholders to describe the exact same ideas: first, we are going to say that this wave has an amplitude of 1 (maximum distance from the x-axis) and second that it has a wavelength of 2 (distance between points where the wave starts to repeat itself). We are now going to see how we can change both the amplitude and the wavelength of y=sinx.

To start, we will change the wavelength of this equation. We will accomplish this by multiplying x before it’s had a chance to pass through the sine function. Think about it this way: whenever we enter 2in, we want the answer we get at --the point twice as far along the curve. We would write this equation out as y=sin(2x), which changes the values from the table above to those below:

x 0 pi/4 pi/2 3pi/4 pi 5pi/4 3pi/2 7pi/4 2pi |
y |

Note how the value for y that we get now at 2 is equivalent to our old value for and now 3pi/4 is equivalent to our old value for 3pi/2. This, then produces the graph below.

Sure enough--our graph still has amplitude 1, but now it begins repeating at rather than 2.

But what about the amplitude? Changing the value of x before it integrates into the sine function overall didn’t have any effect the amplitude of the y values for the equation, so we will have to make this amendment after the sine function has already performed its function. Let’s do this by checking our values for y=2sinx. Now our table comes out with these values:

x 0 pi/4 pi/2 3pi/4 pi 5pi/4 3pi/2 7pi/4 2pi |
y 0 1.414* 2 1.414* 0 -1.414* -2 -1.414* 0 |

* Again, I have truncated a longer number to only 3 decimal places for clarity’s sake.

Now our y values go between -2 and 2 rather than -1 and 1 while the function goes back to repeating at 2. As you can see, adding a 2 to this equation in these different spots produces radically different results! Here is the graph for this equation:

Success!

There is one final transformation we can perform on this equation that does not change the amplitude or the wavelength, but is nonetheless important. By now you may have noticed that the origin of this graph always occurs at 0 on the x axis. There is a relatively simple method we can use to bump this graph a little bit to either side slightly, starting it in a slightly different spot along the curve. To do this, we actually have to add or subtract values rather than multiplying. We can perform this transformation either before or after the sine function performs its duty--the results wind up being the same. For this particular transformation, we can use either y=sin(x+2) or y=sinx+2. Either one will produce the exact same result, but I’m partial to the former approach so that’s what we’ll use for our demonstration. Now our table of values comes out:

x 0 pi/4 pi/2 3pi/4 pi 5pi/4 3pi/2 7pi/4 2pi |
y 1 0.707* 0 -0.707* -1 -0.707* 0 0.707* 1 |

* Yup! Once again these values have been truncated!

Now our equation starts at 1 rather than 0. The resulting graph looks like this:

Now the graph begins on a peak at 0 rather than halfway between a peak and a valley. There’s another fun side-effect of this transformation, as well. You may have noticed we have yet to graph anything with our cosine function--it turns out that y=cosx produces the exact same result as y=sin(x+2). Please note: this will not be the case for most other values we could add to x in this equation--there are a whole range of different starting points available to us for this wave!

With these results in mind, then, we can think of a generalized equation for graphing wave functions that looks something like this:

y=asin(bx+c)

Where a is the amplitude of the wave, b is the wavelength, and c is how far we are displacing the wave along the x axis. Knowing what these three variables do will be hugely helpful in navigating the rest of this paper!

Now let’s apply this knowledge to graphing curves that meet back up upon themselves, such as circles. Because sine and cosine graphs of the same variable have peaks that are offset by a quarter of a wavelength, we can use the two functions together to graph a circle by using the equations x=sint and y=cost where t is equal to all values between 0 and 2.

Why 2? Instead of degrees, mathematicians using trigonometry more frequently use a method for measuring angles called radians. Because the circumference of a circle is equal to double the radius times pi (2pir), we can imagine setting the value of r aside to think of the distance around the circle as being 2. We then can think of any fraction of that distance as being the measure of the angle we are hoping to achieve. For example, 90 degrees is the angle we achieve by dividing a circle into ¼, so we can also express the size of this angle by dividing 2 by 4, where we get a value of 2. If it’s easier to think in degrees than radians, simply take any value of t I present in the rest of the paper and perform the following operation on it to get the equivalent in degrees (d): d=t(180/pi). For the rest of this paper we will assume t = all values between 0 and 2 unless otherwise noted.

**A Note on the Unit Circle**

Using sine and cosine functions to graph a circle is a technique used in geometry called the unit circle--a circle with a radius of 1 unit. The traditional convention is to graph the unit circle and all derivatives of it by assigning the cosine function to the x axis and the sine function to the y axis, giving us x=cost and y=sint. I’m not using the traditional approach to graphing the unit circle for two reasons: the first is that it sets the start point for any given pattern at the far right side of the x axis, leading to shapes like triquetras pointing to the right instead of up and I personally prefer the aesthetics of those patterns pointed up naturally. The second is much more subjective and it’s that my dominant direction of spin is clockwise, whereas the traditional approach to graphing the unit circle leads to the pattern being built counter-clockwise.

If you’re a traditionalist, you can just reverse the equations on the x and y axis to get results that will fit into the traditional convention for graphing curves of this sort. All of the conclusions reached in the course of this paper are valid whether this operation is performed or not.

Tune in tomorrow to learn how to use this type of math to model basic poi flowers!

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