Tangent Space & getting Punched in the Face

I used to think vectors were the same as points. They’re lists of numbers, 
right?
Tuples with each item ∈ the same domain (or same data type). But … they’re 
not.

The way vector mathematics describes the forces of the world is to instantiate 
one

Tangent Space & getting Punched in the Face

I used to think vectors were the same as points. They’re lists of numbers, 
right?
Tuples with each item ∈ the same domain (or same data type). 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.

Bam.

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.

[B9yVHIcXT6IdAAAAAElFTkSuQmCC]

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.

header

That’s football for simple tangent spaces—the Aristotle-v-Newton scenario. 
Now onto
the most poetic sport, where two holonomic dynamical systems dance within each 
other’s
gyre, moving in and out of each other’s movements 
and—occasionally—osculating a
tangent that links one part of me with one part of you.

[7777hallba]Arely v McMorrowoh my god look at this full on uppercut Dunaway 
sneaking
it inclose in uppercut Salcido v Wilcox


All of the logic about connecting vectors head-to-tail with parallelograms is 
only to
reason about one 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.

To talk about striking different points requires a connection (parallel 
transport).
[And remember if you hook my cheek versus jab my forehead, those arenotparallel 
so
even more maths is involved. Also hard tissue (bone) versus soft tissue (cheek).]

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).



Parallel Transport on a Torus - Houdini and Python from Macha on Vimeo.
I'll let you look up "tetrad"

Keeping my torso over the ball is even yet another calculus—one that I suspect 
is
much, much more complicated. 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.

[AUT_0047]

judo counterthrow moving gif

[article13410380C91176A000005DC370_306x423]

[miura]

Did you know Ronda Rousey's mom is a statistician? @annmariastat

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! Back to the base namespace) 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). And sure, you could FEM-approximate my interior with 
a
3-D mesh and a deep geometric logic—and even make the ribs more rigid than the
liver—but that’s just going to make you want higher maths even more as you 
grapple
with the army of ε’s you’ll convoke with all the approximations, dx size, 
parameters,
and so on.

Hehehehehe. Pu jols[baseball-swing][joe-mauer]

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