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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.
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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
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
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
