Projections, Extent, and Resolution

HES 505 Fall 2025: Session 8

Matt Williamson

Describing Absolute Locations

Coordinates: 2 or more measurements that specify location relative to a reference system

Cartesian coordinate system
origin (O) = the point at which both measurement systems intersect
Adaptable to multiple dimensions (e.g. z for altitude)

Locations on a Globe

The earth is not flat…

Latitude and Longitude

Locations on a Globe

The earth is not flat…
Global Reference Systems (GRS)
Graticule: the grid formed by the intersection of longitude and latitude
The graticule is based on an ellipsoid model of earth’s surface and contained in the datum

Global Reference Systems

The datum describes which ellipsoid to use and relies on angular measurements to describe location on a sphere

Geodetic datums (e.g., WGS84): distance from earth’s center of gravity
Local data (e.g., NAD83): better models for local variation in earth’s surface

Global Reference Systems

Great for precise location
Great for worldwide communication
Not great for measurements (e.g., distance, area)
Inconsistent accuracy

Global Reference Systems

The World Is Not Flat

But maps, screens, publications, and Cartesian Coordinates are…
Projections describe how the data should be translated to a flat surface
Rely on ‘developable surfaces’
Strictly: the mathematical function to transform the globe into the developable surface.

Projections

It’s just maths…

\[ \begin{aligned} n &= \tfrac{1}{2}(\sin \phi_1 + \sin \phi_2) \\[4pt] C &= \cos^2 \phi_1 + 2n \sin \phi_1 \\[4pt] \rho(\phi) &= \frac{\sqrt{C - 2n \sin \phi}}{n}, \quad \rho_0 = \frac{\sqrt{C - 2n \sin \phi_0}}{n} \\[8pt] x &= \rho(\phi)\,\sin\!\big(n(\lambda - \lambda_0)\big) \\[4pt] y &= \rho_0 - \rho(\phi)\,\cos\!\big(n(\lambda - \lambda_0)\big) \end{aligned} \]

\[ \begin{aligned} \phi &= \text{latitude} \\[4pt] \lambda &= \text{longitude} \\[4pt] \lambda_0 &= \text{central meridian} \\[4pt] \phi_0 &= \text{latitude of origin} \\[4pt] \phi_1, \phi_2 &= \text{standard parallels} \end{aligned} \]

Coordinate Reference System

Contains the full recipe for translating Earth to coordinates
Begin with a datum (model of spheroid Earth)
Projection - the math for flattening (e.g., Albers, Mercator, etc)
Parameters - the projection center, parallels, etc

Coordinate Reference Systems

Some projections minimize distortion of angle, area, or distance
Others attempt to avoid extreme distortion of any kind
Includes: Datum, ellipsoid, units, and other information (e.g., False Easting, Central Meridian) to further map the projection to the GCS
Not all projections have/require all of the parameters

The Orange Peel Analogy

A datum is the choice of fruit to use. Is the earth an orange, a lemon, a lime, a grapefruit?

A projection is how you peel your orange and then flatten the peel.

Projections are Political

Projection necessarily induces some form of distortion (tearing, compression, or shearing)

Mercator vs. Equal Earth projection from Correct the Map

Projections are Political

Convey power through size
Convey interest through focus
Convey worldview through orientation

Scale and Geographic Analysis

Matching Inquiry to Process

What do we even mean?

Grain: the smallest unit of measurement
Extent: the areal coverage of the measurement

From Manson 2008

Scale

Even if it exists, how do we know we are measuring at the right scale?

Fallacies

Locational Fallacy: Error due to the spatial characterization chosen for elements of study
Atomic Fallacy: Applying conclusions from individuals to entire spatial units
Ecological Fallacy: Applying conclusions from aggregated information to individuals

In Practice

Who chooses extent?
How does aggregation affect the data?
How spatially uniform is the underlying data?
What can we change?

In Practice

Projection is not just a visual choice
Aligning data is critical for robust analysis
How might your choice of distortion affect your analysis

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