From: israel@unixg.ubc.ca (Robert B. Israel)
Newsgroups: sci.math
Subject: Re: Mercator Projection
Date: 9 Nov 92 23:35:41 GMT
Organization: The University of British Columbia
NNTP-Posting-Host: unixg.ubc.ca

In <1992Nov8.214329.27209@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>In article <israel.721212129@unixg.ubc.ca> israel@unixg.ubc.ca (Robert B. Israel) writes:
>>In <a34uTB4w165w@netlink.cts.com> kfree@netlink.cts.com (Kenneth Freeman) writes:
>>
>>>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).

>Turns out if you try to calculate this using the 1986 Encyclopedia
>Britannica you get sec^3(t).  The reason is that EB defines the
>Mercator Projection to be the result of projecting the globe from its
>center onto the cylinder tangent to the equator.  If this were true the
>vertical direction would scale not by sec(t) but by the derivative of
>tan(t), namely sec^2(t).

>Since the Rand McNally Mercator projection of the world hanging in our
>kids' playroom fits your formula exactly, and since the EB definition
>would make Greenland (.84M sq.mi) look at least five times bigger than
>South America (6.8M sq.mi) (it looks roughly the same size), I'd say
>you were right.

Appalling goof by the EB!  I looked in a 1967 Britannica, and they didn't
make that error (although they didn't give the formula either).

>So how is the Mercator projection defined?  One way I've seen is that
>it maps rhumb lines (lines of constant bearing, not sailors waiting for
>their daily ration) to straight lines of the corresponding slope, which
>would seem to determine it uniquely up to dilatation (translation or
>scaling).

This is the feature that made it valuable to 16th century mariners:
you just need to draw a straight line on the map from where you are
to your intended destination, measure the angle between this and a
meridian, and keep that heading on your compass until you get there
(with some complications due to the difference between magnetic north
and true north).

>An immediate consequence of this definition is that the Mercator
>projection is conformal (locally shape-preserving).  However
>conformality in the plane is weaker than dilatation (e.g. z^2 as a
>transformation of the complex plane, which sends z+e to z^2+2ze for
>small e, rotating and scaling e by 2z).  So conformality alone isn't
>enough to define the Mercator projection.  One might ask for the
>projection to be linear, but what does linearity mean when projecting
>from a globe?  The EB definition gives a notion of linear projection
>from a globe, but unfortunately it's wrong.

>So with the goal being to patch the EB definition:

>1.  What is the weakest condition required in addition to conformality
>to uniquely determine the Mercator projection up to dilatation?

If you conformally map the sphere with the North and South poles removed, 
1-1 onto a cylinder, don't you have the Mercator projection up to 
translation and reflection?

>2.  What other natural definitions exist for the Mercator projection?

A conformal map in which all meridians are vertical.

A physical model: blow up a spherical balloon inside a cylinder, and have
it stick to the walls of the cylinder when it touches them.  

>The EB should use the best definition.
>-- 
>Vaughan Pratt

By the way, for a very entertaining exposition of some of the history of
the Mercator projection and its mathematical aspects, see the article
"An Application of Geography to Mathematics: History of the Integral of
the Secant" by V.F. Rickey and P.M. Tuchinsky, Mathematics Magazine
vol. 53 #3 (May 1980), 162-166.  Especially recommended for calculus
instructors trying to motivate the integral of sec(x).
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Newsgroups: sci.math