Oh -- I think I've got it now.

I've spent the past week trying to figure out just what the fuck is happening foot-wise in figure skating jumps. The downstairs neighbors are probably wondering if I've moved a small elephant in, as the house is old and and has wooden flooring that goes whump every time I land on it. You can actually do these things on dry land. Not well, mind you. Earning enough airtime for a double or triple anything requires you to pick up a significant amount of speed. You could probably do an axel with a running start, kind of, but all of the other ones take off going backwards, and you'd need to be on skates where you can turn yourself around quickly without losing momentum.

The closest thing I already have to an axel in my dance repertoire is a pirouette, a front-traveling spin that rotates on one foot and lands on the other. There's even a variation that puts your feet into what skaters call the 'back spin position', where one leg is straight and the trailing leg wraps forward around it, crossing at the ankle. Dancers pull all their limbs in while spinning for the same reason skaters do, which is that a constant linear velocity translates into a larger angular velocity when the accelerated mass moves closer to the axis of spin.

(It also works the other way. Optical readers like CD- and DVD-ROM drives spin at a constant angular velocity -- that is, a set number of revolutions per second -- which translates to a larger linear velocity as you move farther away from the center of the disc. The arc length traversed by the pickup laser over a wedge of N radians on a circle of radius r is [(N/2π) * 2πr] = Nr; if the drive spins through N radians in time t, and the media contains d units of data in a given linear length of track, then the pickup will read at a rate of Nrd/t, which will obviously get bigger as r increases. A 52x CD-ROM drive only actually reads at 52 times the standard transfer rate at the outer edge of the media. Optical discs are read from the inside diameter outwards, so a lot of high-throughput things like games take advantage of this quirk by padding the filesystem with junk or zeroes, forcing the usable data outwards where it will be read faster.)

To perform a comparable pirouette, step out on one foot, transferring all of your weight onto it. Pop the stepped-out foot up onto the ball (or, if you're in toe shoes, en pointe). At the same time, use the trailing foot to push off going backwards -- if you stepped out left, this will make you turn clockwise; if right, counterclockwise -- and pull your arms in towards your body. Relax the knee on your trailing leg, and it will settle naturally into 'backspin' as you turn. Complete one or more rotations; to stop, step out of the spin with the leg you're not standing on.

In a pirouette, the main propulsive force is actually the foot you're balancing on. The trailing leg determines which way you travel, but unless you're making a specific effort to turn or move the spin in a particular direction, you'll end up standing on your stronger, dominant leg. A professional ballerina performing en pointe can can keep the spin going after the initial kick-off simply by flipping her standing foot down flat and hopping back up on her toe again with each revolution. Friction is reduced by supporting yourself on a minimal cross-section -- i.e., on your toes, rather than your whole foot -- and can be reduced further by wearing slippery shoes, which is why pointe shoes are traditionally covered in satin, rather than made of leather/vinyl like ballet slippers and jazz shoes.

The problem with this is, compared to an axel, the feet are backwards. In a pirouette, it's your back push-off foot that ends up wrapped around the now-straightened front leg, and which steps out of the turn in the end by unwrapping; in an axel, your lead foot is supposed to end up crossed in front of the one that swept up from behind, and you come down on the straightened leg, not the wrapped one. A pirouette and an axel also spin opposite directions from a given step-out foot: taking off from your left sends you counterclockwise, and taking off from your right sends you clockwise.

The explanation in Wikipedia was not a help. I find people are just about as terrible at assessing what they're actually doing with their hands and feet as they are at assessing what's coming out of their mouths. Everyone thinks they're perfectly coherent in conversation, only to be appalled when they set eyes on an exact transcript, which is full of ums and uhs and you knows and interrupted and unfinished sentences. The diagram in the article on axel jumps is missing too many I-frames and the path with arrows means just about nothing to me when projected onto a 2D plane, and the text explanation has a lot of annoying lacunae. It's all well and good to tell me that the back foot 'steps up' into the jump and order me to cross my left foot over my right, but how do you step up? Knee bent? Toe pointed? Swing it wide or fling it straight in? And how do I get into a position where it's possible to cross my ankles like that without stabbing myself in the other skate or trying to swing a foot counter to my rotational direction?

The difference is the location of the center point you're theoretically trying to rotate around. In a pirouette, although the spin travels forward, it travels in between rotations. You step forward, and then shift your weight until it's centered directly over the lead foot, which is where the spin part happens. The lead leg stays straight because it's providing all of the upward pushing and because the act of tensing it also carries much of the rotational energy, and therefore the trailing push-off foot has to be the one that wraps around. In an axel, the lead foot is providing only the upwards force, which is stored before takeoff by bending that knee -- the center of the leg is not co-located with the spin axis the way it is in a pirouette, and the forward-traveling vector actually comes from the velocity you'd built up before beginning the jump. Once you've vaulted off the toe pick, that foot abdicates all responsibility for supporting you, and that knee can relax enough to wrap the leg naturally around the other one as you turn.

The leg you "step up" with, on the other hand, needs to stay as long and straight as possible, because the faster you can get that foot swinging around while it's eccentric, the more angular velocity it will translate to when you pull it back to center while in the air. The stepping-in motion is described thus because it mimics stepping down against a floor -- if you were on the ground, you'd be stepping forward and transferring your weight to it with the motion. Relative to your body, the axis of the spin is straight down the center, rather than down one leg; relative to the ice, you're aiming to turn yourself around a point ahead of and above you. The axes are coincident in a pirouette because you halt your forward and upward progress while you turn, whereas in an axel you don't begin rotation until you've flung yourself off your pick in both the X and Y directions.

The sequence is essentially reversed from a pirouette. You start the rotation before you've finished shifting your weight, rather than after. It ends in similar mirror image: You put your foot down so you can brake against the floor to stop your pirouette, but you stop your axels by opening out to bleed off angular velocity so you can safely touch down on the ice.

...this must be what they teach as ballet in Bizarro World.

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