7.2 Projected Coordinate Systems & Map Projections

7.2 Projected Coordinate Systems & Map Projections

While a Geographic Coordinate System (GCS) defines positions on a 3D ellipsoidal model of the Earth using angular units, computer display monitors and physical paper maps are two-dimensional flat surfaces. A Projected Coordinate System (PCS) mathematically converts 3D spherical coordinates (latitude and longitude) into 2D Cartesian coordinates ($X$ and $Y$) on a flat plane. Every Projected Coordinate System is built upon an underlying Geographic Coordinate System and utilizes linear units of measurement (such as meters or feet). Mastering map projections, distortion trade-offs, and standardized projection grids—such as State Plane and UTM—is critical for passing the Esri ArcGIS Pro Foundation certification exam.

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The Structure of a Projected Coordinate System

A Projected Coordinate System consists of two essential components:

1. An underlying Geographic Coordinate System (GCS) (which specifies the ellipsoid, datum, and prime meridian).
2. A Map Projection (a set of mathematical formulas that projects spherical coordinates onto a 2D Cartesian coordinate plane).

Linear Units of Measurement

Unlike a GCS, which measures positions in angular units (decimal degrees), a PCS uses linear units measured from a designated origin point $(0,0)$. Common linear units include:

• Meters (m): The international standard metric unit used in UTM and global projections.
• International Feet (ft): Defined exactly as $1\text{ foot} = 0.3048\text{ meters}$.
• US Survey Feet (ftUS): Historically defined as $1\text{ meter} = 39.37\text{ inches}$ (approximately $1\text{ foot} \approx 0.3048006096\text{ meters}$).

Note for GIS Practice: The minute distinction between US Survey Feet and International Feet amounts to approximately 2 feet per million feet. In high-precision engineering and surveying projects, selecting the wrong foot definition can cause noticeable dataset offsets.

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Developable Projection Surfaces

To visualize how a map projection works, cartographers project features from a transparent 3D globe onto a 2D geometric surface that can be flattened without stretching, tearing, or creasing. These geometric shapes are known as developable surfaces. The three primary developable surfaces are:

1. Cylindrical Projections

A cylinder is wrapped around the globe touching along a line of tangency (typically the Equator) or intersecting the globe along two lines (secant case). When unrolled, the grid of latitude and longitude forms a rectangular grid of parallel straight lines.

• Characteristics: Excellent for global mapping and north-south trending strips.
• Examples: Mercator, Transverse Mercator, Cassini.

2. Conic Projections

A cone is placed over the globe like a hat, touching along a single parallel line of latitude (tangent case) or intersecting along two parallel lines (secant case, known as standard parallels).

• Characteristics: Ideal for mid-latitude regions with broad east-west extents (e.g., the continental United States or Canada).
• Examples: Lambert Conformal Conic, Albers Equal Area Conic, Equidistant Conic.

3. Planar (Azimuthal) Projections

A flat plane touches the globe at a single point (tangent case) or intersects along a circle (secant case). The point of contact is often placed at the North Pole, South Pole, or a specific city.

• Characteristics: Ideal for polar mapping, circular regions, and aviation route planning.
• Examples: Stereographic, Orthographic, Azimuthal Equidistant.

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Distorted Spatial Properties: The Core Cartographic Trade-off

It is mathematically impossible to flatten a 3D curved surface onto a 2D plane without distorting geometric properties (a principle formalized by Carl Friedrich Gauss as Theorema Egregium). Every map projection distorts at least one of four fundamental spatial properties:

Exam Tip: For quantitative spatial analysis measuring total area (such as calculating square kilometers of forest loss), analysts must use an Equal Area projection. For navigation or local survey mapping where local angles and shapes matter, analysts choose a Conformal projection.

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Projection Parameters

Map projections are customized using specific mathematical parameters:

• Central Meridian ($X_0$): The line of longitude chosen as the origin for X-coordinates.
• Latitude of Origin ($Y_0$): The line of latitude chosen as the origin for Y-coordinates.
• Standard Parallels: Lines of latitude where the projection surface intersects or touches the globe. Scale distortion is exactly zero ($1.0$) along standard parallels.
• False Easting: A linear value added to X-coordinates to eliminate negative coordinate values within the mapping extent.
• False Northing: A linear value added to Y-coordinates to eliminate negative coordinate values.

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State Plane Coordinate System (SPCS)

Developed in the 1930s to provide a unified high-accuracy grid for surveying across the United States, the State Plane Coordinate System (SPCS) divides the US into over 120 distinct geographic zones. Each zone uses a customized projection designed to maintain scale distortion below 1 part in 10,000 ($0.01%$).

To achieve minimal distortion, SPCS selects the developable surface based on zone geometry:

• Lambert Conformal Conic: Used for states or zones elongated in an east-to-west direction (e.g., North Carolina, Tennessee, Pennsylvania).
• Transverse Mercator: Used for states or zones elongated in a north-to-south direction (e.g., Illinois, Indiana, Georgia).
• Hotine Oblique Mercator: Used for the Alaska panhandle due to its diagonal alignment.

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Universal Transverse Mercator (UTM)

The Universal Transverse Mercator (UTM) system is a worldwide projected coordinate system covering the Earth from latitude $84^\circ\text{N}$ to $80^\circ\text{S}$.

UTM key parameters:

• Divides the globe into 60 longitudinal zones, each spanning 6 degrees of longitude wide. Zones are numbered 1 through 60, starting at $180^\circ\text{W}$ longitude and moving eastward.
• Uses a Secant Transverse Mercator projection within each 6-degree zone.
• Assigns a False Easting of 500,000 meters to the central meridian of each zone, ensuring that all X-coordinates remain positive.
• In the Northern Hemisphere, the Equator has a False Northing of 0 meters. In the Southern Hemisphere, the Equator is assigned a False Northing of 10,000,000 meters to prevent negative Y-coordinates.