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- Newsgroups: sci.math
- Path: sparky!uunet!snorkelwacker.mit.edu!galois!riesz!jbaez
- From: jbaez@riesz.mit.edu (John C. Baez)
- Subject: Re: Mercator Projection
- Message-ID: <1992Nov10.033657.10730@galois.mit.edu>
- Sender: news@galois.mit.edu
- Nntp-Posting-Host: riesz
- Organization: MIT Department of Mathematics, Cambridge, MA
- References: <a34uTB4w165w@netlink.cts.com> <israel.721212129@unixg.ubc.ca> <1992Nov10.024331.10080@galois.mit.edu>
- Date: Tue, 10 Nov 92 03:36:57 GMT
- Lines: 33
-
- In article <israel.721212129@unixg.ubc.ca> israel@unixg.ubc.ca (Robert
- B. Israe) writes:
- >In <a34uTB4w165w@netlink.cts.com> kfree@netlink.cts.com (Kenneth
- Freeman) writs:
- >
- >>My Mercator projection goes 'up' to only 84 degrees, ~the northern
- >>tip of the classically huge Greenland. I'd like to know three things.
- >>1) For a given area, what is its apparent increase in size for a
- >>given latitude? I.e., what is the rate of increase the closer you
- >>get a pole (and infinity)?
- >
- >At latitude t, linear dimensions are multiplied by sec(t), so areas are
- >multiplied by sec^2(t).
-
- Hmm, maybe I'm confused. Let t be the angle from the equator
- (latitude) and x the distance from the equator on the Mercator
- projection. (The Mercator projection draws a line from the
- center of the earth through the surface of the earth and then to an imaginary
- cylinder running north-south into which the earth fits snugly, right?)
- Then x = sec t, where I'm assuming the earth has unit radius.
- But if you want to know the amount by which linear dimensions
- are multiplied *right at latitude t* you need dx/dt = sec^2 t.
- Areas would then go as sec^4 t. (Here you need to note that
- east-west lengths are getting scaled the same way as north-south
- lengths, i.e. that the Mercator projection is conformal.)
-
- I hope I'm wrong, because if it were really dx/dt = sec t, then we would
- have a reason to figure out the integral of sec t -- a notoriously
- sneaky basic integral. And I even seem to recall some remark about it
- being the Mercator projection that led people to do this integral.
-
- So corrections are welcome.
-
-