Maths IA · Geometry and trigonometry · topic 32 of 40
Map projections: how do they distort the world? · IB Maths IA idea: guide, sample graph and step-by-step plan
HL maths hiding inside a geography questionGreenland looks as big as Africa on many maps. You're going to put numbers on exactly how wrong that is, and why.
- ✓ you like geography and maps
- ✓ you are happy with trigonometry and calculus
- ✓ you want a topic where definitions matter
The big idea
← definitions first, formulas secondYou cannot flatten a sphere without distorting something. A projection is a function from latitude and longitude to the plane, and distortion is measured by scale factors: how much a small length or area is stretched. Define a projection precisely, work out its scale factors, and compare it with another. Clear definitions come first: what is a distance, what is a scale factor, and at which point? Projections are in neither course, so define everything yourself; the HL-level maths is in deriving the scale factors yourself and justifying any formula you borrow.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Start from the condition that small shapes keep their shape (angles preserved) and derive the formula.
→ A real derivation with an integral of sec φ; an exploration of why the poles are infinitely far away.Pair Mercator with a projection that keeps areas correct.
→ A direct comparison of the area scale factors and a discussion of what each projection protects and what it sacrifices.Measure distance on a map and on the sphere using the great-circle formula.
→ A table of true and apparent distances with an error measure, and a link to vectors or the cosine rule on a sphere.Find the area of a region on a sphere by integration, and compare its map area.
→ A real application of integration; the ratio of areas shows the distortion at that latitude.Think of a small circle on the globe and what shape it takes on the map.
→ The scale factors in two directions at that point, and a clear picture to draw.Choose real countries and measure their area on a printed map.
→ Your own data, with uncertainty in the measurement, and a test of how well the model predicts it.Your Maths IA research question: run the check
say WHICH distortion you measureWrite your own question first. Then make sure it ticks every box:
- Names the projections compared and states exactly what each one does to latitude and longitude.
- Defines what 'distortion' means here: a scale factor for lengths, for areas, or both, and at which points.
- Says what quantity you will calculate and what you will compare it with: a published value or a measurement.
- Says how you will deal with the sphere: its radius and units.
- Is narrow enough for 12 to 20 pages, with HL-level maths you can fully explain.
- Can be answered with numbers, not with a statement that 'maps are wrong'.
The picture to draw
A globe, equal small circles on it, and where they land on an unrolled Mercator map. Mercator is not a ray drawing: you derive the height of each latitude. Draw yours for the projections you choose.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Your question; the map or the place that made you ask it.
What you will define, derive and compare.
Latitude, longitude, distances along great circles, a small area on a sphere; the notation you will use throughout.
The function from the sphere to the plane, for each projection, with a clear diagram and every symbol defined.
A derivation of the length and area scale factors by trigonometry and calculus. At HL this is where rigour and sophistication must show.
Tables and a graph of the scale factors against latitude, for two projections.
Real distances or areas measured on a map, compared with the model, with uncertainty and a number for the difference.
What each projection protects and sacrifices; what a mapmaker would choose for what purpose; limits of a spherical Earth.
Answer with numbers, and say whether the aim was met.
Sources, data and extra tables.
What your Maths IA graph could look like
1 = no distortion: say that on the axisMade-up data, to show you the form of a strong graph, not the answer. Yours will look different. example only
- The axis tells the reader what the factor is relative to: 1 means no distortion in area.
- Two curves on the same axes, so the comparison is made by eye as well as by numbers.
- Error bars on both axes: the latitude of a country is not a single number.
- The curve rises very steeply, so say where you stop (here at 80°) and why.
- Another graph to consider: the length scale factor 1 / cos φ against latitude, or the ratio of apparent to true distance for real city pairs.
Your game plan
check one case you know by heartTick steps off as you go. We'll remember on this device.
0 of 11 done
- Read and predictFind what a map projection is and what 'distortion' can mean. Before any calculation, predict how the area of a region changes with latitude.
- Fix your definitionsWrite what a distance, an area and a scale factor are, and at which point you measure them. Clear definitions come first.
- Trial run (one evening)Take a globe and an atlas page. Measure a few distances and compare. Do you see the effect? Do your units agree?
- Choose projections and a questionPick two projections, one simple and one that protects something different. Say why.
- Derive the formulasWork out the projection formulas and the scale factors, step by step. Check one against a known case, such as the equator.
- CompareTabulate and graph the scale factors against latitude for both projections.
- Test with real placesChoose cities or countries with reasons. Measure on a printed map or from a source with a stated scale. Compare with the model by a number.
- Go one step deeperChoose one: area on a sphere by integration, great-circle distance, or the shape of a small circle. Justify each step.
- Reflect as you goWhat does each projection protect and sacrifice? What does the model assume about the Earth?
- Write, then cut12 to 20 pages. One worked derivation per type, then tables. Notation, axis labels and units all need checking.
- Check before your teacher's draftDoes the conclusion answer your aim with numbers? Is every symbol defined once and used the same way throughout?
Maths you'll need
Traps to avoid in this Maths IA
- Vague definitions of 'distortion' and 'scale', so the reader cannot tell what you calculated.
- Using a projection formula from a website without deriving or checking it.
- Treating the Earth as flat, or the radius as unknown, for the distance calculations.
- Measuring areas on a printed map with no scale or uncertainty.
- Forgetting that a scale factor changes with latitude: quoting a single value for the whole map.
- Only describing the projections in words and never calculating anything.
- Too wide: three projections and five cities. Choose two projections and one deeper step.
More Maths IA ideas like this
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.