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