Geospatial Coordinate Systems and Map Projections – Study Notes

Geographic Coordinate Systems (GCS)

• Topo maps and coordinate systems come in unprojected (GCS) and projected forms.
• Unprojected systems are based on spherical globe coordinates; expressed as degrees of latitude and longitude.
• Projected systems convert spherical coordinates to a planar representation using a set of mathematical projection equations, effectively mapping 3D coordinates to a 2D map.

Types of coordinate systems

• Unprojected (GCS): uses degrees of latitude and longitude.
• Projected: converts spherical coordinates to planar coordinates via a projection function.
• Topo maps commonly include three coordinate systems:
  • One unprojected system: Geographic Coordinate System (GCS) in degrees.
  • Two projected systems: State Plane (feet) and UTM (meters).

Example: same point, different x-y representations

• A single geographic location may have different x-y values depending on the coordinate system:
  • GCS (degrees): e.g., 103°21'43.83"W, 44°6'49.26"N.
  • UTM Zone 13 (meters): e.g., Easting ≈ 631,058.40 m, Northing ≈ 4,885,805.77 m.
  • State Plane (feet): e.g., Easting ≈ 1,204,817.08 ft, Northing ≈ 663,391.19 ft.
• This illustrates that coordinates are system-dependent, even for the same physical point.

Spatial reference attributes

• A complete description of a dataset’s coordinate system is required for proper display and analysis:
  • Geographic Coordinate System (GCS) / datum
  • Projection (if used)
  • Storage units for x-y values (degrees, feet, meters, etc.)
  • Domain: maximum allowable x-y values
  • Resolution: x-y precision

Geographic coordinate systems

• Geographers use latitude and longitude to specify positions on the globe.

Measuring degrees

• Latitude: measures the angle from the horizontal, representing north-south distance from the equator.
• Longitude: measures around the circle of the equatorial plane, representing east-west distance from the Prime Meridian.

Ellipsoids

• The Earth is not a perfect sphere; mapping a point onto an ellipsoid provides a better fit for the Earth’s shape.
• An ellipsoid is defined by a major axis (a) and a minor axis (b), representing the longer and shorter radii.
• Ellipsoid parameters have changed over time as measurements of the Earth's shape improved.

The geoid

• The geoid is a theoretical surface defined by gravity; it represents mean sea level extended through the continents.
• The geoid is irregular and complex to map, so the ellipsoid is used as a practical reference surface.
• The geoid is the mean ocean surface in equilibrium; gravity measurements help define it.

Datum

• A datum shifts the ellipsoid relative to the geoid to best fit the geoid for a given area.
• Local datums optimize the fit for a particular location (possibly using a surveyed network of points).
• Geocentric (world-centered) datums optimize fit for the entire Earth.

North American datums

• NAD27 (North American Datum 1927): based on Clarke 1866 ellipsoid; widely used until the 1980s, still in some datasets.
• NAD83 (North American Datum 1983): based on the GRS80 ellipsoid; current popular datum for most mapping; good default when datum is undocumented.
  • NAD83 uses the GRS80 ellipsoid.
• NAD83(HARN): High Accuracy Regional Network updates to NAD83 with fitted points.
• WGS84 (World Geodetic System 1984): Geocentric datum; commonly used as the default for GPS units.

Projections and datums

• Every projection is based on a Geographic Coordinate System (GCS).
• Every GCS has a datum.
• Every projection has a datum as well.
• Projections based on different datums will be offset from one another; the amount of offset depends on the region and is typically between 0 and 300 meters, though it can be larger regionally.
• Example: UTM Zone 13 NAD83 vs NAD27 differ due to the underlying datum.

Projected coordinate systems

• Projected coordinate systems map the Earth’s surface onto a plane using a projection method and a datum.
• The projection can influence how distances, directions, and areas are preserved or distorted.

Projections

• A projection mathematically transforms points from the curved Earth surface to a flat plane.
• The source surface is typically defined by an ellipsoid and a datum.
• Different datums yield slightly different results on the projected plane.

A Cartesian coordinate system (illustration concept)

• Cartesian coordinates express positions as (x, y) in a plane with an origin (0,0).
• Examples show different origin placements and axes directions; fundamental idea is to map 2D space with linear coordinates.

Types of projections

• Cylindrical projections
• Conic projections
• Azimuthal (planar) projections

Cylindrical projections – general idea

• Often used for world maps; generally preserve direction and/or shape depending on the variant.
• Transverse cylindrical projections are good for north-south oriented areas.

Conic projections

• Examples: Lambert Conformal Conic, Equidistant Conic, Albers Equal Area Conic, Polyconic.
• Often used for east-west extents; can preserve area and distance depending on the variant.

Azimuthal projections

• Examples: Lambert Azimuthal Equal-Area (for example at 90°N, 0°W as shown in figures).
• Commonly used for satellite data and polar regions; typically preserve area and distance.

Applications by projection type

• Cylindrical: generally preserves direction and shape (with caveats).
• Transverse Cylindrical: good for north-south oriented areas.
• Conic: generally preserves area and distance; suitable for east-west extents.
• Azimuthal: generally preserves area and distance; good for polar regions and satellite data.

Distortion in projections

• All map projections introduce some distortion.
• The type and degree of distortion vary with the chosen projection.
• When selecting a projection, choose one with properties suitable for the intended analysis (e.g., preserving area, shape, distance, or direction as needed).

Universal Transverse Mercator (UTM) Grid – key features

• The UTM grid divides the world into zones; each zone has:
  • An origin at the zone’s principal meridian (central meridian).
  • A northing reference with 0 meters at the equator for the northern hemisphere.
  • A false easting of 500,000 meters to ensure positive eastings.
  • A stated northing value that can extend up to 10,000,000 meters (false Northing) in the southern hemisphere to avoid negative values.
• Example concept from figures:
  • Mt. Rushmore is approximately
  • Easting: 123{,}733.22 m east of the zone’s principal meridian.
  • Northing: 4{,}859{,}580.26 m north of the equator.
  • Zone example shown as Zone 13 (N).

State Plane Coordinate System (SPCS)

• State Plane zones are regionally defined, typically by state and zone (N/S/E/W divisions).
• Maps show many SPC zones across states (e.g., AK-1, WA-N, PA-N, CA-2, UT-N, etc.).
• SP coordinates are typically in feet (or meters, depending on the zone and datum).
• The mapping in figures includes a grid of zone codes across the U.S., illustrating broad regional coverage.

Map scales

• Map scale definitions:
  • Scale ratio (dimensionless): e.g., 1:24,000.
  • Representative fraction (RF): same idea as a ratio.
  • Graphic scale: a visible bar or line on the map showing ground distance corresponding to map distance.
• Example: 1:24,000 means 1 unit on the map represents 24,000 of the same units on the ground.
• Example: 1 inch on the map equals 2,000 feet on the ground (since 1 inch = 2.54 cm; 24,000 inches ≈ 2000 feet for this scale in the provided example).
• Common note: 1:24,000 is a typical large-scale map used for detailed topographic work; larger scales have more detail.

Large vs small scale

• Large scale: e.g., 1:5,000 – shows more detail; used for things like city blocks, buildings, street networks.
• Small scale: e.g., 1:50,000,000 – shows less detail; used for continental or global views.

Examples of map scales and distances (table-like guidance)

• For common scales, approximate ground distances represented by map distances are:
  • 1:2,000 → 1 cm on map ≈ 20 m on ground
  • 1:5,000 → 1 cm on map ≈ 50 m on ground
  • 1:10,000 → 1 cm on map ≈ 100 m on ground
  • 1:24,000 → 1 inch on map ≈ 2,000 ft on ground
  • 1:50,000 → 500 m on ground per cm
  • 1:100,000 → 1,000 m on ground per cm
  • 1:250,000 → 2,500 m on ground per cm
  • 1:500,000 → 5,000 m on ground per cm
  • 1:1,000,000 → 10,000 m on ground per cm
  • 1:5,000,000 → 50,000 m on ground per cm
  • 1:10,000,000 → 100,000 m on ground per cm
• Additional notes:
  • 1 cm on map represents different real-world distances depending on the unit choice (meters, feet, kilometers, etc.).
  • The same fundamental ratio can be expressed in terms of inches or centimeters depending on map units.

Commonly used map scales (specific examples)

• 1:2,000; 1:5,000; 1:10,000; 1:24,000; 1:25,000; 1:50,000; 1:100,000; 1:125,000; 1:126,720; 1:250,000; 1:500,000; 1:1,000,000; 1:5,000,000; 1:10,000,000
• For quick reference, the following mapping is typical:
  • 1 cm on map ≈ 20 m on ground (1:2,000)
  • 1 cm on map ≈ 50 m on ground (1:5,000)
  • 1 cm on map ≈ 100 m on ground (1:10,000)
  • 1 inch on map ≈ 2,000 ft on ground (1:24,000)
  • 1 cm on map ≈ 500 m on ground (1:50,000)
  • 1 cm on map ≈ 1,000 m on ground (1:100,000)
  • 1 cm on map ≈ 2,500 m on ground (1:250,000)
  • 1 cm on map ≈ 5,000 m on ground (1:500,000)
  • 1 cm on map ≈ 10,000 m on ground (1:1,000,000)
  • 1 cm on map ≈ 50,000 m on ground (1:5,000,000)
  • 1 cm on map ≈ 100,000 m on ground (1:10,000,000)

Specific example: Mt. Rushmore in UTM coordinates

• Mt. Rushmore example coordinates (illustrative from the figures):
  • Easting: approximately 123{,}733.22 m east of the zone’s principal meridian
  • Northing: approximately 4{,}859{,}580.26 m north of the equator
  • Zone: Zone 13 (N) in UTM context

Quick recap: what to keep in mind

• Coordinate systems can be unprojected (degrees) or projected (planar units such as meters/feet).
• A dataset requires a clear description of its spatial reference (GCS, projection, units, domain, resolution).
• Ellipsoids, geoid, and datums are core concepts that determine how coordinates map to the real Earth.
• Projections trade off distortions of area, shape, distance, or direction; no projection preserves everything perfectly.
• UTM and State Plane are two major projected coordinate systems used in North America; each has zones that define their origin and false references to keep coordinates positive and manageable.
• Map scales determine how map distances relate to real-world distances; common scales range from large-scale (detailed) to small-scale (overview).
• Real-world examples (e.g., Mt. Rushmore) illustrate the practical use of easting/northing, central meridians, and zone designations in UTM.