In the previous article, we gave a place two numbers: latitude and longitude.
So the next step seems easy.
Draw those coordinates on a flat sheet of paper.
But Earth is curved. Paper is flat.
Something has to change.
Quick Answer
A round Earth cannot be flattened into one plane without distortion.
A map projection can preserve some useful properties very well, but not all of them everywhere at once.
The usual trade-offs involve: area, shape, distance, and direction.[1]
A map projection is not a perfect copy of Earth. It is a choice about what to preserve and what to let distort.
Try Flattening an Orange Peel
Imagine peeling an orange in one piece.
Now try to press the curved peel flat on a table.
It will wrinkle, tear, stretch, or overlap.
A world map faces the same geometric problem.
The surface of Earth has curvature. A flat map does not.
To move points from one surface to the other, a mathematical rule — a map projection — is needed.
Figure 1. Every world map makes trade-offs. Different projections protect different properties.
What Can a Map Distort?
Four questions are especially useful.
Area:
Does a country look as large, relative to another country, as it really is on Earth?
Shape:
Do small local features keep their angles and form?
Distance:
Does the spacing between two places on the map represent their real separation?
Direction:
Does a direction drawn on the map correspond to the direction we want to use on Earth?
USGS summarizes the problem simply: a flat map may show one or more of these properties well, but never all of them everywhere on the entire Earth.[1]
A Simple Clue: Longitude Lines Converge
Look at a globe.
Lines of longitude are far apart at the equator. They move closer together as they approach the poles.
On a simple rectangular latitude-longitude grid, those same longitude lines may be drawn with equal spacing from left to right.
Something must be stretched.
On a spherical teaching model, the east-west distance represented by one degree of longitude is approximately proportional to:
cos(latitude)
At the equator, that factor is 1.
At 60° latitude, it is only 0.5.
Near the poles, it approaches zero.
The globe itself is telling us that one rigid rectangular scale cannot stay correct everywhere.
Python — Watch East-West Distance Shrink
We can make that idea visible with a few lines of Python.
import math
km_per_degree_at_equator = 111.2
for latitude in [0, 30, 60, 80]:
distance = (
km_per_degree_at_equator
* math.cos(math.radians(latitude))
)
print(latitude, round(distance, 1), "km")You should get values close to:
| Latitude | Approx. km per 1° longitude |
|---|---|
| 0° | 111.2 km |
| 30° | 96.3 km |
| 60° | 55.6 km |
| 80° | 19.3 km |
The numbers shrink because longitude lines converge toward the poles.
Predict first.
Add 89 degrees to the list.
What should happen to the east-west distance as you approach the pole?
Ptolemy Already Faced the Projection Problem
This problem is much older than modern GIS.
Ptolemy's Geography did not only list locations by latitude and longitude. It also discussed ways of representing the spherical world on a plane.[2]
That tells us something important.
Once people began describing Earth with coordinates, they immediately faced the next mathematical question:
How should a curved coordinate grid be drawn on a flat surface?
Mercator Chose Navigation
Many centuries later, Gerardus Mercator faced a practical problem from navigation.
A sailor may want to keep a constant compass bearing. On a globe, that path is generally a curve called a rhumb line or loxodrome.
Mercator's 1569 world map used a projection in which such constant-bearing paths could appear as straight lines.[3]
That was extremely useful for navigation.
But the benefit came with a cost.
Mercator is a conformal projection: it preserves local angles and therefore small local shapes well. But area and distance distortion grow strongly toward the poles.[1]
Mercator was not a bad map because it distorted area. It was a map built to protect something else: useful direction and local angle relationships.
Another Map Can Make a Different Choice
Suppose your question is not navigation.
Suppose you want to compare the true relative sizes of countries, forests, or climate zones.
Then an equal-area projection may be more useful.
Equal-area projections preserve area relationships, but shapes must distort somewhere.[1]
An equidistant projection makes selected distances correct, but cannot preserve every distance between every pair of points.[4]
This is why there is no single “best world map.”
There is only a better projection for a particular purpose.
This Is an Engineering Decision
That last sentence is bigger than cartography.
In engineering, we often cannot optimize every desirable property at the same time.
We choose what matters most, understand what is sacrificed, and make the trade-off visible.
Choosing a projection works the same way.
purpose → choose what matters → accept distortion elsewhere
Where This Lives Today
Digital maps still use projections.
Web mapping commonly uses Web Mercator because it works conveniently for interactive tiled maps, but measurements of distance or area should not automatically be trusted in that projected view.[5]
GIS professionals choose coordinate systems and projections according to the geographic region and the property that matters for the task.[6]
The map on a screen may look effortless.
Behind it is a mathematical design choice.
Words to Keep
map projection
A mathematical rule for representing locations on a curved Earth on a flat surface.
distortion
A change in area, shape, distance, or direction introduced by a map projection.
conformal
A projection property that preserves local angles and small local shapes.
equal-area
A projection property that preserves relative area.
trade-off
A choice in which improving one property requires accepting a cost in another.
One Sentence to Keep
Every flat world map distorts something, so the right projection depends on what we need the map to do.
What Should We Ask Next?
Cartographers faced one version of a fascinating problem: how do we put a three-dimensional world onto a two-dimensional surface?
Renaissance artists faced another version.
They wanted a flat painting to look like deep space.
Our next question is:
How Did Renaissance Artists Put 3D Space on a Flat Page?
Previous: How Did People Turn the Earth into a Grid?
Sources & Further Reading
- U.S. Geological Survey, “How are different map projections used?” — concise explanation of direction, distance, area, and shape trade-offs.
- MacTutor History of Mathematics, “Ptolemy” — Ptolemy's geographical coordinates and work on projecting spherical surfaces onto a plane.
- MacTutor History of Mathematics, “Cartography” — Mercator's 1569 projection, rhumb lines, and the navigation problem.
- U.S. Geological Survey, Guide to Selecting Map Projections — properties and limitations of equal-area, equidistant, and directional projections.
- Esri Support, “Why are my map, distance and area measurements wrong when using WGS 1984 Web Mercator?” — practical explanation of Web Mercator's area and distance distortion.
- Esri, “Select a Suitable Map Projection or Coordinate System” — choose a projection according to the property most important for the task.
Mathematical note: the cosine calculation uses a spherical Earth as a teaching model. Real geodetic work uses an ellipsoidal Earth model and more precise formulas.