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 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
Homolosine Projection
Van der Grinten
Bonne
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