Projections and Coordinates
Vital Resources
• John P. Snyder, 1987, Map Projections – A
Working Manual, USGS Professional Paper
1395
• To deal with the mathematics of map
projections, you need to know trigonometry,
logarithms, and radian angle measure
• Advanced projection methods involve calculus
Shape of the World
• The earth is flattened along its polar axis by
1/298
• We approximate the shape of the earth as an
ellipsoid
• Ellipsoid used for a given map is called a
datum
• Ideal sea-level shape of world is called the
geoid
Shape of the World
• Earth with topography
• Geoid: Ideal sea-level shape of the earth
– Eliminate topography but keep the gravity
– Gravity is what determines orbits and leveling of
survey instruments
– How do we know where the sea would be at some
point inland?
• Datum: Ellipsoid that best fits the geoid
• Sphere: Globes and simple projections
The Datum
Datums
• In mapping, datums is the plural (bad Latin)
• Regional datums are used to fit the regional
curve of the earth
– May not be useful for whole earth
• Obsolete datums often needed to work with
older maps or maintain continuity
Regional Datum
The Geoid
Distortion
• You cannot project a curved earth onto a flat
surface without distortion
• You can project the earth so that certain
properties are projected without distortion
– Local shapes and angles
– Distance along selected directions
– Direction from a central point
– Area
• A property projected without distortion is
preserved
Preservation
•
•
•
•
Local Shape or Angles: Conformal
Direction from central point: Azimuthal
Area: Equal Area
The price you pay is distortion of other
quantities
• Compromise projections don’t preserve any
quantities exactly but they present several
reasonably well
Projections
• Very few map “projections” are true projections
that can be made by shining a light through a
globe (Mercator is not)
• Projection = Mathematical transformation
• Many projections approximate earth with a
surface that can be flattened
– Plane
– Cone
– Cylinder
• Complex projections not based on simple
surfaces
Choice of Projections
• For small areas almost all projections are
pretty accurate
• Principal issues
– Optimizing accuracy for legal uses
– Fitting sheets for larger coverage
• Many projections are suitable only for global
use
Projection Surfaces
Simple Projection Methods
Orthographic Projection
Gnomonic
Butterfly Projection
Dymaxion Projection
Azimuthal Equal Area
Azimuthal Equal Area
Azimuthal Equidistant
Stereographic
Equirectangular (Geographic)
Equirectangular
Projection
Mercator
Transverse
Mercator
Oblique Mercator
Lambert Equal Area Cylindrical
Peters Projection
Ptolemy’s Conic
Lambert Conformal Conic
Albers Equal Area Conic
Polyconic Projection
Bipolar
Oblique Conic
Mollweide
Aitoff Projection
Sinusoidal
Robinson
Mollweide Interrupted
Mollweide Interrupted
Homolosine Projection
Van der Grinten
Bonne
Specialized Projection
Specialized Projection
Transverse Mercator Projection
UTM Zones
UTM Pole to Pole
Halfway to the Pole
USA Congressional Surveys
Grid vs. No Grid
Wisconsin
Grid
Systems