Map Projections and Coordinate Reference Systems
Why every flat map distorts Earth, and how to choose coordinates that fit the question
Before You Start
You should know
That latitude and longitude are a way of naming location on Earth.
You will learn
What a coordinate reference system is, why map projections always involve distortion, and how to choose coordinates that match the calculation you want to do.
Why this matters
Distance, area, slope, buffering, and overlay can all go wrong if the coordinate system is wrong. A projection choice is not a cosmetic setting. It is part of the model.
If this gets hard, focus on…
One simple idea: Earth is curved, but most maps and most computations are flat. A projection is the bridge between those two facts.
In school atlases, Greenland often looks almost as large as Africa. It is not. Africa is about fourteen times larger. The map is not lying exactly, but it is distorting. That distortion is the price of taking a curved planet and flattening it onto paper or a screen.
Computational geography runs into this problem constantly. We measure river lengths, buffer roads, calculate slope from elevation grids, compare forest area, and interpolate environmental fields. Every one of those tasks depends on coordinates. If the coordinates do not match the geometry of the question, the numbers can become misleading before any interesting modelling even begins.
This chapter gives that problem a proper foundation. It explains what a coordinate reference system is, why projections always distort something, and how to choose a sensible working system for the analysis in front of you.
A CRS Is A Chain Of Decisions, Not Just A Label In A Dropdown
Readers often meet coordinate systems as software settings, which hides the real structure. A useful mental model is a short chain: Earth has a shape, the datum ties coordinates to that shape, the projection flattens it if needed, and the final choice should match the measurement question rather than map appearance alone.
Start With Curved Earth
Longitude and latitude name position on a curved planet, so they are good for global storage and exchange.
- angular units
- great for GPS and interchange
- awkward for direct metre-based analysis
Add Datum + Projection
The datum decides which mathematical Earth is being used. The projection decides how curved positions become flat coordinates.
- datum controls alignment
- projection creates x-y map coordinates
- every flattening introduces distortion
Match The Task
The right CRS depends on what you are measuring, not on what looks familiar.
- distance/buffers: local projected
- area totals: equal-area
- display/browsing: web map CRS may be fine
Geographic CRS
Use longitude-latitude when the goal is global referencing, data exchange, or preserving one common Earth-wide location format.
Best question: “Where is it on Earth?”
Projected CRS
Switch to metre-based coordinates when buffering, measuring distance, calculating slope, or doing local spatial analysis.
Best question: “How big, how far, or how steep?”
Property-Specific CRS
Choose equal-area, conformal, or other families based on which property needs the most protection.
Best question: “What must distort least?”
1. The Question
How do we put locations from a curved Earth onto a flat map without confusing ourselves or breaking the analysis?
That question has three parts:
- How do we name location on Earth?
- How do we flatten a curved surface into a map?
- How do we decide whether a given coordinate system is suitable for distance, area, or direction calculations?
If you can answer those three questions, you can avoid one of the most common failures in GIS and spatial modelling: using coordinates that look plausible but measure the wrong thing.
2. The Core Idea
A Coordinate System Is Not Yet A Full CRS
A coordinate system tells you how positions are written.
Examples:
- geographic coordinates: longitude and latitude, often written as (\lambda, \phi)
- planar coordinates: eastings and northings, often written as (x, y)
But a full coordinate reference system, or CRS, does more than name axes. It also tells you:
- what shape of Earth is being assumed
- where the origin and orientation come from
- how curved positions are transformed into flat-map positions, if a projection is involved
So when someone says “the CRS,” they mean the whole positioning framework, not just the coordinate labels.
Earth Is Curved, Analysis Is Usually Flat
Earth is approximately spherical, and more accurately ellipsoidal. A map on a page or screen is flat. You cannot move from one to the other without distortion.
An orange-peel analogy is useful here:
- leave the peel curved and it fits the orange
- flatten the peel and it tears, stretches, or compresses
Map projections do the same thing mathematically. They preserve some properties better than others, but they cannot preserve everything at once.
The Four Big Properties
Projection choices usually trade among four properties:
- area: are regions shown with correct relative size?
- shape: are local angles and outlines preserved?
- distance: are measured lengths reliable?
- direction: are bearings preserved from some point or along some lines?
No single projection preserves all four everywhere.
That is why projection choice depends on the question:
- area comparison needs an equal-area projection
- local shape and angle work often prefers a conformal projection
- short-distance engineering work often uses a local projected CRS such as UTM
- global overview maps may prioritize visual continuity over precise measurement
Read this as a decision picture, not as a universal scoreboard. Equal-area projections protect size comparisons, conformal projections protect local shape, and local projected systems such as UTM are often the most convenient working surface for ordinary distance-based analysis.
3. The Building Blocks Of A CRS
Geographic Coordinates
Geographic coordinates use:
- longitude \lambda: east-west angle
- latitude \phi: north-south angle
These are angular coordinates, measured in degrees, not metres.
That point matters. A degree of longitude is not a fixed ground distance. It shrinks as you move toward the poles.
Datum
A datum tells you how the coordinate system is tied to Earth.
In plain language, a datum answers: what mathematical Earth are we measuring on, and how is that mathematical surface aligned with the real planet?
You do not need geodesy in full detail here. The important practical point is this:
- two datasets can both use latitude and longitude
- but if they use different datums, they may not line up exactly
For most modern work, you will often see datums such as WGS84 or NAD83.
Projection
A projection converts positions on the curved Earth into positions on a flat surface.
That means:
(\lambda, \phi) \longrightarrow (x, y)
You do not need the full formula for every projection. What matters first is that the output coordinates are now planar and usually measured in metres.
EPSG Codes
Many CRSs are identified by an EPSG code, which is a standard reference number.
Examples:
EPSG:4326for geographic coordinates in WGS84EPSG:32612for UTM Zone 12N in WGS84
You do not need to memorize the numbers. You do need to recognize that an EPSG code is a compact way of saying, “use this exact CRS definition.”
4. Geographic Versus Projected Coordinates
Geographic Coordinates Are Good For Storage And Exchange
Latitude and longitude are excellent for:
- global data exchange
- GPS output
- storing location on the whole planet
- mapping very large extents in a common reference
But they are awkward for direct measurement because the units are angles, not metres.
Projected Coordinates Are Better For Many Calculations
Projected coordinates are usually better for:
- buffering by a fixed distance
- measuring length
- measuring local area
- terrain derivatives that depend on horizontal spacing
- overlay and engineering-style site analysis
This is why many workflows begin in geographic coordinates, then reproject into a projected CRS for analysis.
Worked Example: Why Degrees Are Not Uniform
A rough rule is:
- 1 degree of latitude is about
111 km - 1 degree of longitude is about
111 cos(\phi) km
At the equator, where \phi = 0^\circ:
111 \cos(0^\circ) = 111 \text{ km}
At latitude 53^\circ, roughly the latitude of Edmonton:
111 \cos(53^\circ) \approx 111 \times 0.60 \approx 67 \text{ km}
So one degree of longitude is:
- about
111 kmat the equator - about
67 kmnear Edmonton
That is why a “square” one-degree cell is not really square on the ground, and why area and distance work in raw longitude-latitude coordinates can quickly become misleading.
This is the visual reason geographic coordinates are awkward for direct measurement. The east-west size of a degree changes continuously with latitude, so the same numeric difference in longitude does not mean the same ground distance in different places.
5. Common Projection Families
UTM
The Universal Transverse Mercator system divides Earth into zones and gives each zone a local projected grid.
UTM is often a strong default for:
- local to regional analysis
- distance measurement
- buffering
- terrain work
Why it is useful:
- units are metres
- local distortion is small inside a zone
- it is easy to reason about eastings and northings
Equal-Area Projections
Equal-area projections preserve area correctly.
They are useful when you care about:
- land-cover totals
- watershed area
- habitat comparison
- regional statistics by polygon
If the question is “which region covers more land?” an equal-area CRS is often the safest choice.
Conformal Projections
Conformal projections preserve local angles and local shape reasonably well.
They are useful for:
- navigation
- local orientation work
- some engineering and topographic mapping tasks
But preserving local shape does not mean preserving area.
Geographic CRS
A geographic CRS is still a valid CRS. It is just usually not the best working CRS for planar measurement.
Think of it as a good storage and interchange format, but often a poor analysis surface for operations that assume metres.
Named Examples You Will Actually See
Some projection names appear so often that they are worth recognizing directly.
Mercator
- Preserves local shape reasonably well
- Great for marine navigation and web-map familiarity
- Strongly inflates high-latitude area
- Poor choice for comparing region size
Sinusoidal
- Equal-area projection
- Good for global rasters and land-cover accounting
- Shapes stretch away from the central meridian
- Useful when area matters more than visual familiarity
Albers Equal-Area Conic
- Equal-area projection built for mid-latitude regions that extend east-west
- Common choice for national and provincial area comparison
- Often a strong fit for Canada-wide thematic mapping
- Better for area totals than for preserving local shape perfectly
Lambert Conformal Conic
- Conformal projection commonly used for aeronautical and topographic mapping
- Strong for mid-latitude regions with large east-west extent
- Preserves local shape and angle better than area
- A very plausible choice when navigation, orientation, or regional basemapping matters
Polar Stereographic
- Designed for polar and high-latitude mapping
- Useful when Arctic geometry matters more than equatorial balance
- Common in sea ice, cryosphere, and polar remote-sensing products
- Distortion grows away from the pole, but it treats the Arctic far more honestly than Mercator
Azimuthal Equidistant
- Preserves distance and direction from a chosen center point
- Useful for “how far from this city, station, or radar?” questions
- Not a general-purpose equal-everything projection
- Best when one origin point is the star of the map
UTM
- Local projected grid in metres
- Excellent for regional measurement, buffering, and terrain work
- Not designed for whole-continent or global mapping in a single zone
- Best read as a practical working CRS rather than a world-view map
Same Earth, different flattening choices. These sketches are not exact cartographic outputs. They are teaching diagrams that show how the graticule changes shape under several common choices.
These examples point to a practical habit: when someone names a projection, ask what it is trying to preserve and what scale it was designed for. "Mercator," "Sinusoidal," "Albers," "Lambert conformal conic," and "Polar stereographic" are not just labels; they are answers to different cartographic priorities.
### A Canada And Alberta Decision Example
The projection question becomes much easier once you attach it to a real place and a real task.
Suppose we are working in western Canada:
- Edmonton is around `53.5°N`
- Calgary is around `51.0°N`
- northern Alberta reaches much farther poleward
- the province is large enough that distortion choices matter, but small enough that we usually do not need a whole-world map
Now imagine three different jobs:
**Job 1: Buffering wells, roads, or streams around Edmonton**
This is a local measurement task. We care about metres. A local projected CRS such as the relevant UTM zone is usually the right working surface.
**Job 2: Comparing burned area or wetland area across all of Alberta**
This is an area-comparison task. A provincial or national equal-area projection is usually safer, because northern Alberta should not get an artificial size bonus from the map.
**Job 3: Showing Alberta in a web map with familiar zoom and pan behavior**
This is mostly a display task. Web Mercator may be acceptable for visual browsing, but it is still a poor surface for area comparison.
So the question is not "which projection is best?" The better question is "best for what?"
```{=html}
<div data-viz="echarts" style="height:420px; min-height:320px" data-options='{
"title": {"text": "Choosing a Projection for Alberta and Western Canada", "subtext": "Different spatial questions call for different working surfaces", "left": "center"},
"tooltip": {"trigger": "axis", "axisPointer": {"type": "shadow"}},
"legend": {"bottom": 0, "data": ["Local projected (for example UTM)", "Equal-area", "Web Mercator"]},
"grid": {"left": "7%", "right": "4%", "bottom": "16%", "containLabel": true},
"xAxis": {
"type": "category",
"data": ["Local distance work", "Provincial area comparison", "Terrain derivatives", "Public web display"],
"axisLabel": {"interval": 0, "rotate": 12}
},
"yAxis": {"type": "value", "name": "Suitability (teaching scale)", "min": 0, "max": 5},
"series": [
{
"name": "Local projected (for example UTM)",
"type": "bar",
"data": [5, 3, 5, 3],
"itemStyle": {"color": "#2563eb"}
},
{
"name": "Equal-area",
"type": "bar",
"data": [3, 5, 3, 2],
"itemStyle": {"color": "#0f766e"}
},
{
"name": "Web Mercator",
"type": "bar",
"data": [1, 1, 1, 5],
"itemStyle": {"color": "#dc2626"}
}
]
}'></div>This chart is deliberately practical. In Alberta, a projected CRS in metres is usually the working choice for local analysis, an equal-area projection is often the safer choice for province-wide area accounting, and Web Mercator is mostly a display convenience rather than an analysis surface.
One more implication matters for Canada in particular: high latitude exaggeration is not a side issue. It is exactly why Mercator-style views can mislead northern comparisons. If your question is about size, extent, or area totals in northern Canada, projection choice is part of the evidence, not just part of the map styling.
6. What Goes Wrong When The CRS Is Wrong
Buffering In Degrees
Suppose you want a 500 m buffer around a stream.
If the layer is in longitude-latitude and you accidentally buffer by 500, the software may interpret that as 500 degrees, which is absurd. Even if you use a small degree value that seems to approximate 500 m, the ground distance will vary with latitude.
Area Comparison In A Distorted Projection
Suppose you want to compare forest loss in northern Alberta and southern Alberta.
If the projection inflates northern areas more strongly, the comparison can be biased by the map itself, not by the land-cover change.
Slope On A Geographic Grid
Suppose a digital elevation model stores heights on a longitude-latitude grid.
Slope depends on:
\text{slope} \approx \frac{\text{vertical change}}{\text{horizontal distance}}
If the horizontal spacing is treated as though degrees were uniform metres, the slope estimate is wrong. This is why terrain chapters keep warning you to check the CRS before computing derivatives.
The bars show the problem directly: the same degree-based east-west step is almost three times wider at the equator than at high latitude. A projected CRS with metre units removes that ambiguity and makes the buffer distance mean what you think it means.
7. A Practical Selection Habit
When choosing a CRS, ask:
- Is my main task storage, display, or measurement?
- Do I care most about area, distance, shape, or direction?
- Is the study area local, regional, continental, or global?
- Do all layers already share a CRS?
- Are the output units appropriate for the calculation?
A useful beginner checklist is:
- use geographic coordinates for global reference and exchange
- use UTM or another local projected CRS for local distance work
- use equal-area projections for area comparison
- reproject all layers to a common CRS before overlay and measurement
8. Worked Example By Hand
Problem: A road centerline is stored in a projected CRS with metre units. Two points on the road are:
- A = (500000,\ 5910000)
- B = (500600,\ 5910800)
Estimate the straight-line distance between them.
Solution
The coordinate differences are:
\Delta x = 500600 - 500000 = 600 \text{ m}
\Delta y = 5910800 - 5910000 = 800 \text{ m}
Use the Pythagorean formula:
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
d = \sqrt{600^2 + 800^2} = \sqrt{360000 + 640000} = \sqrt{1000000} = 1000 \text{ m}
So the straight-line distance is 1 km.
This is the kind of calculation projected coordinates are designed to support. The numbers behave like ordinary planar geometry because the CRS has already done the work of flattening the Earth in a controlled way.
9. If This Gets Hard, Focus On
- a CRS is the full positioning framework, not just axis names
- latitude and longitude are angles, not metres
- every flat map distorts something
- projected coordinates are often better for measurement
- choose the CRS to match the question, not just the default setting
That is enough foundation for the raster, terrain, vector, and remote-sensing chapters that follow.