Coordinates and Projections
Objectives of this lecture:
- Spatial measurements (coordinates)
- Special problems posed by the Planet Earth
- Flattening the Earth: Projections
The previous lecture dealt with the attributes and how they
are measured. This deals with the spatial component of mapping,
and how the Earth gets onto a flat map.
Coordinates:
pairs
of measurements on orthogonal axes (flat model of Euclid's
Geometry) [illustration courtesy of Peter Dana]
[ developed by Rene Descartes] the lifeblood of digital cartography
Mathematics work neatly- analytical geometry to calculate distance,
direction, etc.
But the earth ain't flat.
"Geodetic" coordinates
- Latitude (sun angle at noon) and Longitude
(time at local noon)
- basic starting point: Axis of rotation of the
earth (North South);
- Equator is plane at right angles to axis; mid-way
between Poles
- Latitude measured as angle from Equator toward Pole;
- "Parallels" of latitude are parallel to
each other; only Equator is a great circle.
- Longitude: relative to (arbitrary) Prime Meridian
(Greenwich)
- time of day gives "meridian" (line with
a common local noon) at right angles to parallels
Ellipsoid: smooth surface generated by rotation an ellipse
around its minor axis (polar axis)
The polar axis is shorter due to dynamic effects of rotation on
the Equator.
Common ellipsoids used for cartography in USA:
Clarke's 1866 : major axis: 6378206.4 meters; flattening:
1/294.98
World Geodetic Reference System (GRS) 1980 : major axis 6378137
meters; 1/298.26
The Basic Problem in a projection:
Represent the Geoid [actual shape of earth - putative sea level]
(simplified into ellipsoid [a surface created by rotation of best
fit ellipse to geoid] or spheroid) on a flat map.
Various forms of distortion are inevitable, but you get
to choose type and amount.
Properties to preserve:
· area preserved - equal area (EQUIVALENT)
· preserve angles - CONFORMAL
distances preserved from a single point - EQUIDISTANT
preserve compass directions (LOXODROMES)
preserve great circles as straight lines (gnomonic projection)
BOTTOM LINE: You MUST pick only one of the above...
Developable Surfaces:
- Analytical projections (not on a simple surface)
Two examples
Mercator projection
Cylindric, conformal, also preserves compass directions
Mollweide projection
Analytical (pseudo cylindric), equivalent (preserves area)
Resources to learn more about the topic:
Coordinate
system tutorial at Geographer's Craft Project
Projection
Tutorial, also at Geographer's Craft
If you liked this lecture... More coming in Lecture
13.
Version of 3 April 2003
