Projections II

Objectives of this lecture:

1. More about Geodesy
2. Projections: a tour beyond those mentioned earlier, common coordinate systems
3. Error in projections: Tissot's Indicatrix
4. EVENT 13.

GEODESY - the science of earth measurement

Properties of spherical geometry:

• Great Circles: a line on Earth made by a plane that passes through the center of the Earth (shortest distance between two points on surface)
• small circles: parallels (except for Equator) are NOT shortest path

Actual shape of the Earth:

Ellipsoid: a smooth surface that approximates the shape of the Earth (see Lecture 03)

Geoid (local gravity surface)

Gravity varies (oceanic rock is denser than continental rock: land floats...)
Thus, "down" isn't always towards the center of the earth exactly. This effect complicates measurement on the surface of the earth. Geodetic surveying attempts to correct the lumpy geoid to the smoothness of an ellipsoid to make numerical calculations possible.

Datum

(odd word: plural is "datums")
An agreed-upon package of ellipsoid plus established values for specific locations
Latuitude - longitude pairs are relative to some known locations (officially Greenwich Observatory for Zero Longitude by international agreement, but in the pre-satellite era, there was no way to make observations across the ocean)

NAD 27 (North American Datum of 1927)

NAD 83

The Basic Problem in a projection: (reminder)

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.

Developable Surfaces:

Projections "map" the 3D solid onto a plane. Geometric construction methods had to use geometric tricks to do this - three primary developable surfaces (and then some more options).

Planar

• ORTHOGRAPHIC ("perspective" view from parallel lines (infinite distance))
• STEREOGRAPHIC (from opposite side of earth) Preserves shape
• GNOMONIC (from center of earth) Great circles are straight lines (seismic/radio waves)

Cylindrical

• RECTANGULAR (each degree is the same...)
• MERCATOR Preserves loxodromes (straight compass bearings)
• MILLER (modified MERCATOR to include poles)
• PETERS / GALL'S EQUIVALENT - Preserves area (makes stringy shapes)

Conic (one or two standard parallels)

• ALBERS Preserves area (in limited areas, shape not badly distorted)
• Lambert's Conformal Conic: Preserves shapes (in limited use, area not badly distorted)

Mixtures of developable surfaces:

• Goode's Homolosine (MERCATOR plus SINUSOIDAL)

Mathematical Projections (not on a surface)

• SINUSOIDAL Preserves area for large portions of globe, radical shape distortions...
• HYPERELLIPTICAL General projection, many possible results...
• Molleweide (special case of HYPERELLIPTICAL)

Each projection can be in a given orientation (or "aspect")

Aspects

• Normal (equator as horizontal line; polar axis as "vertical")

• Transverse

• Oblique

Interrupted projections provide a loop-hole for whole earth projections

Properties to preserve:

Which projections should you remember?

• latitude/longitude graticule meets everywhere at right angles: Cylindric in Normal Aspect
• parallels curve as concentric circles: Conic in Normal Aspect

• Area of Greenland is not shown true to scale relative to Africa on Mercator
• Conformal maps sacrifice area to preserve shape
• Equivalent projections sacrifice shape to preserve area...

Common coordinate systems:

UTM Universal Transverse Mercator Grid (developed by military) 6° zones running pole to pole
State Plane Coordinates (used by engineers)

Other Countries:

Tissot's Indicatrix: Measure of distortion in a projection

creates a circle on the earth (ellipsoid) shows change onto projected map.
Types of distortion: changes in SIZE = scale variation (does not preserve area)
changes in shape from circle = angle variation (does not preserve shape)

Resources to learn more about the topic:

Hunter College Map Projection pages (not updated since 97, but still useful)

How to choose a projection (student discussion)

A reading on distortion

A Gallery of Map Projections (MICROCAM) - navigate to Tissot's Indicatrix, and look at the slide show.

Geographer's Craft Project Dana's Coordinate Systems pages have a Geodetic Datums section and a Projections section

A 1944 paper on the classification of projections (9 page .pdf)