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From: rassilon@eddie.MIT.EDU (Brian Preble)
Newsgroups: comp.graphics
Subject: Re: Mercator projections
Message-ID: <6201@eddie.MIT.EDU>
Date: Fri, 26-Jun-87 23:13:28 EDT
Article-I.D.: eddie.6201
Posted: Fri Jun 26 23:13:28 1987
Date-Received: Sat, 27-Jun-87 12:35:51 EDT
References: <4800@milano.UUCP>
Reply-To: rassilon@eddie.MIT.EDU (Brian Preble)
Organization: MIT, Center for Transportation Studies, Cambridge, MA
Lines: 56

I would like to thank everyone who replied to my original query regarding
mercator projections.  In case anyone is interested, the following is the
best description I received:

	From: seismo!sun!pixar!ph (Paul Heckbert)

	Some people use the term "mercator" rather generically for any
	cylindrical projection, but if you mean the real mercator projection
	then read on:

	A mercator projection is a cylindrical projection which is conformal
	(it preserves the shape of small objects.  That is, an infinitesimal
	square on the sphere maps to an infinitesimal square in the mercator
	projection, whether it's at the equator or near the poles (but not at
	the pole itself).  Therefore latitudes must be stretched proportional to
	the stretching of longitudes.  A parallel at latitude "lat" has length
	2*pi*r*cos(lat) so the longitude stretch factor to make this parallel
	the same length as the equator (to warp the sphere into a cylinder) is
	1/cos(lat) = sec(lat).  We must stretch the latitudes proportionately to
	preserve shape, so we use calculus to get:

		dy = sec(lat) d(lat)  -or-  y = integral[ sec(lat) d(lat) ]

	Where y is the vertical coordinate in the mercator projection.
	The integral turns out to equal:

		y = log(tan(lat)+sec(lat)) = log(tan(lat/2+pi/4)) =
		    log((1+sin(lat))/(1-sin(lat)))/2

	This is sometimes called the inverse gudermannian function.

	So the transform is:
		longitude/latitude coordinates:	(lon,lat)
		mercator coordinates: (x,y)

		x = lon
		y = log(tan(lat/2+pi/4))

	Note that the north and south poles map to infinity.

	Paul Heckbert
	Pixar				415-499-3600
	P.O. Box 13719			UUCP: {sun,ucbvax}!pixar!ph
	San Rafael, CA 94913		ARPA: ph%pixar.uucp@ucbvax.berkeley.edu


Also, for those interested, I wil post a mercator projected map of the U.S.
with some 2700 points following this article.  Once again, thanks to everyone
for your help.

					-- Rassilon
--
Brian Preble
Center for Transportation Studies
Massachusetts Institute of Technology
(617)253-2716