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Published byKelly Watts Modified over 2 years ago

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Geometry of Projections Philip Flip Kromer

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

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

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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.

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

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

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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.

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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!)

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

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

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

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

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

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Fuller Projection

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

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Now States are correct size by population!

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

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World Population 2006

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World Population 2050

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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.

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

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