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.

from Mercator The series begins with the Mercator projection, as at the left.

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.

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.

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.

Trouble? If the movie won't play for you, you may need a more robust media player. (Windows Media Player is notoriously fragile.) One good freeware player is VLC.

Last updated September 24, 2012