Power pull tracings = sine wave?

OK, so here's a question for the science people out there.

Whenever I warm up with my power pulls on a clean sheet of ice, I am visually treated to my tracing that looks like a lovely sine wave. I'd really like to know the physics behind it. E.g., what equations of motion dictate that a power pull tracing is a sine function? Is it really sine or another function?

I know there're lots of physics-y things out there that explain jumps and spins using moment of inertia etc etc. But I don't know anything to explain power pulls. Would appreciate any insight or references!

(Yeah, I'm a math geek... sine functions are as beautiful to me as Mao or Kostner's skating...)

More geek fun : certain kinds of cardioid type shapes defined using polar coordinates look just like loop figures:

http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/Maddox/Maddox.11/Maddox.11.html

Wouldn't your personal optimal shape for efficiency purposes depend on how strong and flexible each of your muscles and ligaments are, along each point of their range of motion?

I'm pretty sure that as a general rule, optimal efficiency and judged figure skating quality are almost orthogonal. We are supposed to do things because they meet the aesthetic standards of figure skating, or because our coach tells us to, not because they are efficient.

cw_skater said:

More geek fun : certain kinds of cardioid type shapes defined using polar coordinates look just like loop figures:

http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/Maddox/Maddox.11/Maddox.11.html

Click to expand...

Interesting that you bring up polar coordinates. (They are actually really cool, no pun intended!) I don't think it's a coincidence that many of the polar equations define graphs that look very much like figure loops and 3 turns. I'm curious to understand if there's a deeper connection. Take for example the simplest polar equation, radius=constant, a circle. We learnt about centripetal forces in high school physics, where a rigid body (e.g. a skater holding an edge) having the right velocity and force will travel in a perfect circle. Voila, there's a figure 8! Of course, skaters aren't rigid bodies, so are capable of much much more interesting patterns of motion.

Query said:

Wouldn't your personal optimal shape for efficiency purposes depend on how strong and flexible each of your muscles and ligaments are, along each point of their range of motion?

Click to expand...

I absolutely agree that individual physique, strength and flexibility plays a role. And may be why in the age of school figures, skaters who excelled at figures didn't do so well at free skating, and vice versa. However, I am working on the assumption that there is a "typical pattern" of a well-executed power pull/figure loop/etc., and the idealized pattern can be described by a certain mathematical equation.

I'm pretty sure that as a general rule, optimal efficiency and judged figure skating quality are almost orthogonal. We are supposed to do things because they meet the aesthetic standards of figure skating, or because our coach tells us to, not because they are efficient.

Click to expand...

I think that to some extent, efficiency is beauty in itself. But that's a discussion for another day.

Naively, I thought the sine-curve like tracings are due to the alternating inside and outside edges. But would be super interesting if there are deeper mathematics behind all of this!

It's been a looong time since I've studied trigonometry.

By sine wave, do you mean the general shape of alternating curves on either side of a long axis?

That's inherent in the nature of repeatedly changing between inside and outside edges, as lishazard says.

Or do you mean the exact shape of the curves, and the relationship between their amplitude and frequency, that you produce when you execute this skill? That's going to vary somewhat depending on each skater's technique.

gkelly said:

It's been a looong time since I've studied trigonometry.

By sine wave, do you mean the general shape of alternating curves on either side of a long axis?

That's inherent in the nature of repeatedly changing between inside and outside edges, as lishazard says.

Or do you mean the exact shape of the curves, and the relationship between their amplitude and frequency, that you produce when you execute this skill? That's going to vary somewhat depending on each skater's technique.

Click to expand...

Yes, I do literally mean sin(x). Or sin(2pi*f*x) if you want to specify the frequency. Maybe (I'm making this up) if the skater is moving at unit speed down the long axis of the rink, and her up/down knee action is like a spring oscillating with some frequency and amplitude, then the displacement of the tracing from the axis will also have the same frequency and possibly different amplitude. Maybe (still making this up) if the knee action is like a perfect undamped oscillator, then the skater's vertical displacement goes like A*sin(2pi*f*x), and the tracing to has horizontal displacement A1*sin(2pi*f*x).

Lol... what I just wrote sounds like completely bogus physics. But I hope I'm getting across the idea of what I'm trying to ask.

I'm going to be thinking about springs next time I warm up my power pulls...