Morphomap
This
movie
exhibits a family of maps with circular-arc
lines of latitude and longitude.
Conceived by the German mathematician Lambert in 1772
and elaborated by his Italian/French colleague Lagrange in 1779,
the family morphs smoothly between two familiar,
but quite different, map forms.
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from Mercator
The series begins
with the Mercator projection, as at the left.
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through Lagrange
As the series evolves, the lines of longitude bulge
outward into circular arcs. Halfway through the series, the
outer arcs become semicircles. This aspect was
originally presented by Lambert, but is usually
singled out as the "Lagrange projection",
probably because several other projections
bear Lambert's name.
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to stereographic
The series ends with the
stereographic projection,
which looks wild here, with the map extended
to a large part of the globe; but the projection is perfectly familiar
when confined to its traditional
role of depicting a hemisphere, as outlined
in pink.
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Technicalities.
In the movie,
the edges bend ever further, from straight at Mercator to
semicircles at Lagrange, and finally become essentially full, infinitely
large, circles at stereographic.
All maps in the family are
conformal, which means they show true shape in every
small area, but with widely varying scale.
(Conformality fails at the poles, except in the
stereographic projection, which is conformal everywhere.)
In the movie
the frames have been adjusted to the same
scale at the center.
Though this makes some maps too big to fit on
the page, only the two end-point maps actually
extend to infinity.
Lagrange identified a further degree of flexibility in the family.
The parallels of latitude can be squeezed
together around one pole and stretched apart around the other.
The movie does not exhibit this extra degree of
freedom. It accords equal treatment to both poles.