Re: Area of Sphere - History

On Wed, 6 Feb 2008 11:06:07 +0000 (UTC), Dave Seaman
<dseaman@xxxxxxxxxxxx> wrote:

The areas are equal. The proof is by integration.

Or to put it another way, it is by looking at a very small area
around a (non-polar!) point on the sphere's surface, and noting
that the tangent plane at the point makes the same angle with the
equatorial plane as the line joining the point on the surface to
the centre of the sphere makes with the polar axis ... (By this
point, I'm desperate to draw a diagram or use symbols. You've got
to admire the ancient Greeks for doing so much geometry in words!)

But the idea is that horizontally (i.e. parallel to the equator)
the projection multiplies lengths locally by R/r, where R is
the radius of the sphere (and cylinder), and r is the distance
from the point on the surface to the polar axis; and vertically,
it multiplies lengths by r/R (because of the two angles being
the same); so areas are unaffected. And since this is true
locally (i.e. at every point), it is (loosely speaking) also
true globally, i.e. when you "integrate".

Is this projection ever used for map-making? Googling for
"equal-area projection" returns a lot of results, e.g.
<http://en.wikipedia.org/wiki/Map_projection#Equal-area>
and it would take a while to search through all of those ...

--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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