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