From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Mercator Projection
Organization: Computer Science Department,  Stanford University.
Date: Wed, 11 Nov 1992 23:16:38 GMT

In article <1992Nov8.214329.27209@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>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?
>
>2.  What other natural definitions exist for the Mercator projection?
>
>The EB should use the best definition.

The responses having slowed down, its about the right time for a
summary.

The best answer to 1 seems to be, nothing else once you've specified
that the projection omits the poles and is onto a cylinder (I hadn't
realized the cylinder requirement would rule out my z^2
counterexample).  The condition that the meridians be vertical is
equivalent to this in the presence of conformality, but is strictly
stronger in its absence, preventing helical meridians for example.  For
2 there is the above; rhumb lines mapped to straight lines of the
corresponding slope; and Robert Israel's balloon model (I finally
convinced myself of its correctness, proof below).

Thanks very much to

John Baez       jbaez@riesz.mit.edu		cylindrical requirement
Kenneth Freeman kfree@netlink.cts.com		for starting this
John Harper     John.Harper@vuw.ac.nz		vertical meridians
Robert Israel   israel@math.ubc.ca		all 3 answers
Alan Paeth      awpaeth@watpix.uwaterloo.ca	much info: taylor(gd(y)) etc.
Ruth Radetsky   vpmath1@sfsuvax1.sfsu.edu	cylindrical requirement
Dennis Ritchie  dmr@research.att.com		Doug McIlroy's C function

The responses combine nicely to give the following articles, written
with both technical and nontechnical readers in mind and thus
complementing Alan Paeth's straight technical writeup.  The first
corrects the misleading Encyclopedia Britannica article without
lengthening it, the second is longer, with no particular application in
mind.

================
1986 article:

Mercator projection, type of map projection develop by Gerardus
Mercator.  Projections are made from the centre of a globe representing
the Earth onto a cylinder surrounding and touching the globe it [sic]
at the Equator; thus, the meridians are equally spaced, parallel
vertical lines, and the parallels of latitude are parallel, horizontal
straight lines, spaced farther and farther apart as their distance from
the Equator increases.

================
Suggested revision:

Mercator projection, type of map projection develop by Gerardus
Mercator.  It is that projection of the globe onto the plane taking
lines of constant bearing to straight lines of that slope, useful in
navigation.  Or, that projection of the globe less its poles onto a
cylinder preserving small shapes: islands appear as seen from the air.
Scale increases as 1/cosine(latitude), distorting larger shapes and
magnifying Greenland tenfold.

================
Longer article:

The Mercator projection is one of a number of projections of a sphere
such as the Earth onto a cylinder, which can then be cut and unrolled
to form a map.  Developed by the Flemish cartographer Gerardus Mercator
in 1569 as an aid to navigation, it has two essential properties.  A
straight line on the map represents a *rhumb line* on the globe, a
course whose bearing is constant and equal to the slope of the line.
And the shape of small regions is depicted faithfully.

The first property permits navigators to plot a course between two
points on the map by connecting them with a straight line, measuring
its slope, and setting a course whose bearing with respect to true
North is that slope.  A drawback is that the shortest distance between
two points on the sphere is not a rhumb line but a great circle or
geodesic, which the Mercator projection renders as a curve attracted to
the nearer pole.  Voyages navigated by rhumb line can be usefully
shortened by breaking them into several rhumb lines connecting points
on a great circle.  Today's navigation equipment makes light work of
exact great circle navigation.

The second property ensures that regions up to a few hundred miles
across appear essentially as they would from a satellite overhead.  The
shape of larger regions however is distorted due to an increase in
scale at higher latitudes, in proportion to sec(L), the secant or
reciprocal cosine of the latitude.  This distortion gives Greenland's
0.84 million square miles a greater apparent area than South America at
6.8 million square miles.

The Mercator projection is the only projection of the sphere less two
antipodal points onto a cylinder having either of these properties.  It
does not however project the sphere radially outwards onto a cylinder
tangent to the equator, as occasionally asserted.  This latter
projection scales nonuniformly, with the vertical scale increasing as
the square of the horizontal scale.  Such a map of the earth would
elongate Greenland to four times its Mercator height without changing
its width, and a course plotted as a straight line on this map would be
both hard to navigate and longer than a rhumb line.

Latitude L on the Mercator projection appears at vertical position
ln(tan(L/2+pi/4)) on the map, the integral of the scaling function
sec(L).  Its inverse, giving latitude from vertical position on the
map, is called the gudermannian function.

A spherical balloon blown up inside a glass tube of diameter that of
the barely inflated balloon will press itself to the sides in exact
accordance with the Mercator projection, until the hemispheres at each
end are stretched to their limit.

======================
In article <israel.721352141> israel@unixg.ubc.ca (Robert B. Israel) writes:
>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.

In article <1992Nov9> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes
>I bet that turns out not to be conformal.  The poor thing's already
>doing its best to stretch horizontally and you're asking it to stretch
>the same amount vertically as well!  Not a chance.

I was able to contradict myself with the following proof.  At any point
the upper portion of the balloon not touching the cylinder must form a
hemisphere with equator a circle in the same state as the undistorted
balloon at that pressure.  (A C1 discontinuity in the balloon shape at
this circle would require a corresonding pressure discontinuity to
maintain it.)  The whole hemisphere having been scaled conformally, the
next increment to stick to the wall will do so conformally.

======================
Dennis Ritchie contributed the following extract from Doug McIlroy's map
program.  struct {float l,s;} pairs angle l in radians with its sine s.
The formula is yet another of the inverse gudermannian's many disguises.
RAD = M_PI/180.

mercator(struct place *place, float *x, float *y)
{
        if(fabs(place->nlat.l) > 80.*RAD)
                return(-1);
        *x = -place->wlon.l;
        *y = 0.5*log((1+place->nlat.s)/(1-place->nlat.s));
        return(1);
}
-- 
Vaughan Pratt
Newsgroups: sci.math