Tangent Space

isomorphismes:

I used to think vectors were the same as points. They’re lists of numbers, right? Tuples. But … they’re not.

The way vector mathematics describes the forces of the world is to instantiate one linear-algebra at each point on the surface where a force is imparted.

When I strike a ball with my foot, look at the exact contact point (and pretend the ball doesn’t squish across my foot—even though it does, because my pegas are so poderosas). There’s a tangent space at that contact point and the exact particulars of my posture, shape of my foot|shoe, and so on determine the force vector that I use to bend the ball like Beckham.

Newton’s f=ma corrected Aristotle’s theory of motion

Vacuum isn’t possible: Vacuum doesn’t occur, but hypothetically, terrestrial motion in a vacuum would be indefinitely fast.

So we know we only need that one strike and then inertia minus drag convolved with lay-of-the-land will determine a full path for the ball: till my teammate’s head contacts it—another tangent space, another vector—and we score.

In any case, all of the logic of connecting vectors head-to-tail with parallelograms is only to reason about that single strike. The whole linear algebra on force vectors is a complete examination of the possibilities of the variations and the ways to strike at that exact same point.

Parallel Transport on a Torus - Houdini and Python from Macha on Vimeo.

Striking the ball on the side (to spin it) or under (to chip it) would be a connection on the S² manifold of the ball — moving the point of tangency — which is a different algebra’s worth of logic. Landing a punch on a different part of me is also a connection (that would be parallel transport of the strike vector on the surface manifold (with boundary) of my face/torso/armpit/whatever).

Keeping my torso over the ball is much, much more complicated—and even though Julian James Faraway and others work on these questions of whole-body mechanics, I don’t think ∃ a complete mathematical theory of how all of the fixed holonomic parts of bodies that have very similar morphogenetic shapes (similar ratios of forearm to humerus length, etc) interact—how soft tissue and hard tissue in the usual places interact and specific angles can make a judoist with excellent technique and little strength able to whirl the body of a weight-lifter around her fulcrum.

Or how this guy can deliver a lot of force with proper dynamical posture (“shape”) when he’s clearly weak and fat. I can start to imagine the beginnings of something like that but it doesn’t obviously fit into the tangent space points & vectors story, except in a very complicated way of vectors connected to vectors connected to vectors, with each connection (not the same as the parallel-transport connection sense of the word I used above!) holonomically constrained or even “soft-constrained” in the case of soft tissue. Same with a blow landing parallel-transported different places on the surface of my body. The transport takes care of the initial strike vector but not how those forces travel and twist through my skeleton, soft tissues, down to the floor (either through my feet or through my *rse if the blow knocks me down).

Or why it’s better to swing the bat this way, or teach your body to do its swimming strokes in that motion, or various other dynamical or static particulars of human body shape and internal anatomical facts.

BTW, an amateur ultimate fighter once told me that the order of importance for winning a fight is:

1. technique (brasilian jiu jitsu technique)
2. balance
3. agility
4. strength

He didn’t mention tolerance for getting punched in the head but he seemed to put up with it remarkably. Also: judo guys have sexy bodies.

(via cab1729)

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