### louder and funnier

here’s the page i munged yesterday, four times bigger
(twice in each dimension, duh). one (more) plainly
sees here that points of an arbitrary hemisphere
(in the top drawing of the lower right panel)
can be identified with longitude-and-latitude pairs
$\langle \phi, \psi \rangle$.

such co-ordinates depend on “choosing
a central meridian” (this was done for
usual planetary co-ordinates by passing
the central meridian through greenwich).
on the drawing (and the planet), we’re
*given* a cutting-of-the-sphere. but
on a more abstractly-given sphere
(in some other context) we might need
to consider *how* the sphere is to be
cut into hemispheres (and *which* hemi-
sphere is to be “drawn” [or what have
you… “studied”]).

on our globe-of-the-world model, this would
amount to selecting a different “north pole”
(notice that this gives us an “equator”,
since the (actual) north pole has a *fixed*
position, we might consider shifting our
“point-of-view” as if we were some satellite,
far enough away to see half the surface
of the “planet” we want to co-ordinatize.

when we look down *from the north pole*,
the “equator” for this point-of-view
is then (of course) the *actual* equator
(the great circle equidistance from the
two poles).

when our satellite is over some *other*
“point” (of our beloved mother earth),
what we’ll see “looking down” is essentially
a *polar projection* of (half of) the surface.
our co-ordinate frame on this model would then
appear as (1) a collection of concentric circles
(the analogues of “latitude”… the angle measured
“up north” [on the north pole model] having been
replaced by the “distance in” [toward the new center])
and (2) a collection of radii (the “meridians”…
measuring the “angle from greenwich” on the
globe and the “angle from the top of the
camera view” (say) in the space-travel version.

all this is perfectly straightforward
& beginners pick it up in a few minutes
of lecture if they’ve got any experience
with maps-of-the-world.

the tricky bits are the reason for all the
the rest of the page. and about these,
i propose to say but little. today.

heck, very little. “identify antipodes.”
here’s a song while i fix breakfast.
ana ng at yootube. (ad-ridden, natch.)

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• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)