Geometry of Projections Philip Flip Kromer

Flatland We communicate in 2d: But the world isn’t 2- (or even 3-) dimensional:

Projection A perfect projection would preserve Distance (isometric) Shape (conformal) Area (equivalent)

Projection A perfect projection would preserve Distance (isometric) Angles (conformal) Area (equivalent) Can’t do this! If we could, a sphere’s geometry would obey Euclid’s axioms.

Think about “un”projecting the map back onto the globe.

Identify “points” and “lines” on globe with image of lines from plane

But a sphere “wraps around”: Hammer a spike at some point in the plane and the same point on sphere. Now put a circle around that point and try to “remove” it.

But a sphere “wraps around” Hammer a spike at some point in the plane and the same point on sphere. Now put a circle around that point and try to “remove” it. On plane, you can’t shrink loop to a point without passing through spike; On sphere, you can do it (go out the other side!)

Projection A perfect projection would preserve Distance (isometric) Shape (conformal) Area (equivalent)

Hammer Projection Not conformal: circles become ellipses, and meridians are curved. However, Area is preserved.

Area Distortion Equatorial Mercator Preserves lines, angles but not area.

Area Distortion Oblique Mercator Distorts distance, shape, and area.

Fuller Projection Don’t need it to be smooth, continuous mapping

Fuller Projection

Weighted Areas Sometimes a good projection is not at all smooth, equivalent, conformal, or isometric 2004 US Presidential Election

Now States are correct size by population!

County size indicates population: - lots of distortion - But demographics clearer

World Population 2006

World Population 2050

Cartogram creation How? Old method: Divide map into cells Scale cells to match population “Fix” edges of neighboring cells to average Diffusion Note that in a finished cartogram, Population density is uniform (why?) Allow population to “flow” until uniform density condition is met.

Diffusion Note that in a finished cartogram, Population density is uniform (why?) Allow population to “flow” until uniform density condition is met.