My first impression was the same as yours, Gareth. But I don't think points follow straight lines in this scheme. With Wright's description, any point on the (unit) sphere will first reach the cylinder when its latitude circle has expanded to radius = 1. By uniform expansion, the local magnification factor in the vertical direction will have to be the same, i.e. 1/cos(lat) = sec(lat). The local equality of magnification factors occurs if and only if angles are preserved. Just considering what happens to a longitude arc A starting from the equator and parametrized by angle t from the horizontal: This arc is mapped by call it t |-> L(t) to the generator of the cylinder meeting the equator at the same point as A, parametrized by height z. By what we know of the magnification, L'(t) = sec(t) (as Gene mentioned), with initial condition L(0) = 0. Thus L(t) = Integral_{0 to t} sec(u) du = ln(sec(t) + tan(t)) Which some people prefer to write as L(t) = ln(tan(t/2 + pi/4). So evidently, contrary to intuition, as the balloon expands and the points reaching the cylinder stop, it moves away from the equator faster than it would from expansion alone. --Dan On Jun 6, 2014, at 6:01 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 07/06/2014 00:15, Dan Asimov wrote:
The first rigorous description of the Mercator projection was apparently given by Edwin Wright in 1599 -- it's a lovely geometric description I hadn't heard before: Imagine a vertical cylinder tangent to the globe at the equator. Now inflate (uniformly expand) a perfectly spherical balloon that initially coincides with the globe, stopping each point at the moment it reaches the cylinder. The correspondence between a point on the globe and the point it reaches on the cylinder is the Mercator projection, and can readily be seen to be conformal.
I'm not sure I understand this.
The simplest interpretation of "uniformly expand" would seem to be that each point moves radially outwards, but that gives exactly the interpretation you said the Encyclopaedia Britannica wrongly gave for years: central projection from the sphere to the cylinder.
So maybe you intend some fancier (more physically realistic?) notion of uniform expansion, but I can't tell what.
What am I missing?
-- g
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