Download Cartography: Map projejctions

Cartography: Map projejctions
We study the geometry of (and determine distances on) the surface
of the earth.
A map projection is a systematic representation of all or part of the
surface of the earth on a plane. This comprises lines called Meridians
(longitudes) and parallels (latitudes).
Since the sphere cannot be flattened onto a plane without distortion,
the strategy is to use an intermediate surface (developable surface)
that can be flattened.
The sphere is first projected onto this surface, which is then laid out
as a plane.
Commonly used surfaces: cylinder, the cone and the plane itself.
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The basic idea for map projection:
consider a sphere with coordinates (λ, φ) for longitude and latitude,
and construct a coordinate system (x, y) so that
x = f (λ, φ), y = g(λ, φ)
where f and g are appropriate functions to be determined depending
on the properties we want our map to possess.
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Equal-area maps. Use to display areal-referenced data. An
example is a sinusoidal projection.
Sinusoidal projection
Equal-area projection.
Obtained by specifying
δg/δφ = R
(R radius of the earth) this imposes equally spaced straight lines for
parallels, and results in the following analytical expressions for f and
g
f (λ, φ) = Rλ cos(φ)
g(λ, φ) = Rφ
Both the Equator and central meridian are standard lines, thus the
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whole map is twice wide as tall.
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Another popular equal-area projection (with equally spaced straight
lines for the meridians) is the Lambert cylindrical projection given by
f (λ, φ) = Rλ
g(λ, φ) = R sin(φ)
This projection’s perspective is easily visualized by rolling a flexible
sheet around the globe and projecting each point horizontally onto
the tube so formed. In other words, light rays shoot from the
cylinder’s axis towards its surface, which is afterwards cut along a
meridian and unrolled.
Like most cylindrical projections, it is quite acceptable for the
tropics, but practically useless at polar regions, which are rather
compressed, resulting in a map much broader than tall. Again like in
other cylindrical projections, deformation is uniform along the same
parallel.
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Properties:
• Meridians are equally spaced
• Parallels get closer near poles.
• Parallels are sines.
• True scale at equator.
• History
– Invented in 1772 by Johann Heinrich Lambert with along
with 6 other projections.
– Prototype for Behrmann and other modified cylindrical
equal-area projections.
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Figure 1: Schematic development of Lambert’s equal-area cylindrical
projection. With a tangent cylinder.
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Mercator Projection
f (λ, φ) = Rλ
g(λ, φ) = R ln tan(π/4 + φ/2)
The great Flemish cartographer Gerhard Kremer became famous
with the Latinized name Gerardus Mercator. A revolutionary
invention, the cylindrical projection bearing his name has a
remarkable property: any straight line between two points is a
loxodrome, or line of constant course on the sphere. In other words,
if one draws a straight line connecting a journey’s starting and
ending points on a Mercator map, that line’s slope yields the journey
direction, and keeping a constant bearing is enough to get to one’s
destination.
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Figure 2: Conventional (equatorial) Mercator map; graticule spacing
10?; map arbitrarily clipped at parallels 85 deg. N and 85 deg. S.
Properties:
• Conformal: it is a projection for which local (infinitesimal) angles
on a sphere are mapped to the same angles in the projection.
• Parallels unequally spaced, distance increases away from equator
directly proportional to increasing scale.
• Loxodromes or rhumb lines are straight. (rhumbs are curves that
intersect the meridians at a constant angle)
• Used for navigation and regions near equator.
• History
– Invented in 1569 by Gerardus Mercator (Flanders) graphically.
– Standard for maritime mapping in the 17th and 18th
centuries.
– Used for mapping the world/oceans/equatorial regions in 19th
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century.
– Used for mapping the world/U.S. Coastal and Geodetic
Survey/other planets in 20th century.
– Much criticism recently.
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Northings and Eastings
Map projections lead to complex equations relating longitude and
latitude to the positions of points on a given map. Thus, rectangular
grids have been developed, in which each point is designated merely
by its distance from two perpendicular axes on a flat map.
The y-axis is the chosen central meridian, y increasing north, and the
x-axis is perpendicular to the y-axis at a latitude of origing on the
central meridian, with x increasing east. The x coordinates are called
eastings and the y coordinates northings. The grid lines do not
coincide with any meridians and parllels except for the central
meridian and the equator.
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Universal Transverse Mercator Projection (UTM)
The world is divided into 60 north-south zones, each covering a strip
six degress wide in longitude. These zones are numbered
consecutively beginning with Zone 1, between 180 degrees and 174
degrees west longitude, going eastward to zone 60 between 174 and
180 degrees east longitude. The northing values are measured
continuously from zero at the Equator, in a northerly direction. to
avoid engative numbers for location south of the Equator, we assigne
the Equator an arbitrary false northing value of 10,000,000 meters. A
central meridian through the middle of each 6 degree zone is assigned
an easting value of 500,000.
The northing of a point is the value of the nearest UTM grid line
south of it plus its distance north of that line; its easting is the value
of the nearest UTM grid line west of it plus its distance east of that
line.
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The UTM system was introduced in the 1940’s by the U.S. Army. It
is widely used in topographic and military mapping.
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Spatial modeling of point-level data often requires computing
distances between points on the earth’s surface. Thus, we can wonder
about a planar map projection, which would preserve distances
between points.
The existence of a map is precluded by Gauss’ Theorem Eggregium
in differential geometrial. Projections perserve area and shapes,
distances are always distorted.
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Calculating distance on the earth’s surface
We must account for the curvature of the earth when computing
distances.
We find the shortest distance (geodesic) between two points.
P1 = (θ1 , λ1 ) and P2 = (θ2 , λ2 ). The solution is
D = Rφ
where R is the radius of the earth and φ is the angle (measured in
radians) satisfying
cosφ = sin θ1 sin θ2 + cos θ1 cos θ2 cos(λ1 − λ2 ).
The geodesic is the arc of the great circle ( a circle with radius equal
to the radius of the earth) joining the two points.
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