Geometry of Projections

Philip Flip Kromer

FlatlandWe communicate in 2d:

But the world isn’t 2- (or even 3-) dimensional:

ProjectionA perfect projection would preserve• Distance (isometric)• Shape (conformal)• Area (equivalent)

ProjectionA 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.

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!)

ProjectionA perfect projection would preserve• Distance (isometric)• Shape (conformal)• Area (equivalent)

Hammer ProjectionNot conformal: circles becomeellipses, and meridians are curved. However,Area is preserved.

Area DistortionEquatorial MercatorPreserves lines, angles but not area.

Area DistortionOblique MercatorDistorts distance, shape, and area.

Fuller ProjectionDon’t need it to be smooth, continuous

mapping

Fuller Projection

Weighted AreasSometimes 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 creationHow?• 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.