Maths IA · Geometry and trigonometry · topic 60 of 97
Why do long flights curve? Great circles · IB Maths IA idea: guide, sample graph and step-by-step plan
curved on the map, straight on the globeOn a flat map, the shortest way from Asia to Europe looks like a straight line. On the globe it isn't, and the flight that looks curved is the shorter one. How much shorter, and where does it matter most?
- ✓ you have ever wondered why a flight map shows a curve and not a straight line
- ✓ you like turning a real-world picture into coordinates and vectors
- ✓ you want maths you can check against a published flight distance
The big idea
Latitude and longitude are angles, and a sphere is three-dimensional. Write each city as a unit vector from the centre of the Earth, and the angle between two cities comes from their scalar product. Multiply that angle in radians by the radius and you have the great-circle distance: the shortest path on the surface. Then ask what a straight line on a flat map really is. A constant-bearing route is easy to fly, but it is longer, and how much longer depends on latitude and on how far east or west you travel. The distance formula is routine. The marks come from what you do with it: derive it in two ways, test it against a published distance, and explain every gap between your answer and a real flight.
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 →
Get the angle between two cities from the scalar product of unit vectors, and again from a chord length and the cosine rule.
→ Two derivations of one result. Show they agree and say which is easier to use for very short distances.Work out the length of a constant-bearing route between the same cities and compare it with the great circle.
→ Needs a map projection, which you define yourself. The percentage difference becomes a function of latitude.Use the haversine form on two cities a few kilometres apart and compare it with the plain cosine form.
→ Rounding errors in a calculator or spreadsheet, and why a mathematically equal formula can be less reliable.Find the highest latitude a great circle reaches between two cities and where it crosses a given meridian.
→ A plane through the centre of the Earth and an optimisation. Check the result on a map or globe.Compare your distances with published flight distances or tracked routes for several city pairs.
→ Winds, airspace and safety rules make real routes differ. Turning those gaps into numbers is your evaluation.Replace the sphere with a slightly flattened ellipsoid, or choose another world with a different radius.
→ Tests how sensitive the answer is to the model. A small percentage change, or a surprisingly large one?Your Maths IA research question: run the check
which cities? which number?Write your own question first. Then make sure it ticks every box:
- Names the places: the city pairs, with coordinates and where they came from.
- Names the quantity you will find: a distance, a percentage extra, or the highest latitude of a route.
- Says what the model is tested against: a published distance or a tracked flight, and what counts as agreement.
- States the Earth model you use (sphere radius, or an ellipsoid) and what you leave out.
- Goes beyond plugging into a formula: there is a derivation to show and a comparison to explain.
The model
Two cities on a sphere, joined to the centre. The distance is the length of an arc, and the angle between the two radii controls it. Draw yours with the cities 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.
Why these routes, and what you expect before any calculation: which is shorter, and by how much?
Latitude and longitude defined, the unit vector for a point derived, the angle between two points from a scalar product.
Arc length from the angle, checked against a second method and against a published distance.
What constant bearing means, how you define the flat map and how you work out its length.
The extra distance for several latitudes and east-west spans, with one graph.
Your distances beside real routes. A number for each difference and a reason for it.
Back to your aim, with the numbers: how much shorter, and where it matters most.
Sphere against ellipsoid, rounding, winds and airspace. Say which assumption matters most.
Where every coordinate and distance came from.
What your Maths IA graph could look like
mark one pair you checked by handA made-up set of readings to show the form of a strong graph, not the answer: the extra length of a constant-bearing route over a great circle, against how far apart two cities are in longitude. Yours will look different. example only
- Model and data on one set of axes, with the model drawn as a curve and your city pairs as points.
- Both axes carry units: degrees of longitude and percent.
- Say what is held fixed (here a latitude of 50 degrees) and test a second latitude.
- Points close to the curve are evidence that your formula and your coordinates are right.
- Another graph to consider: the great-circle distance against the angle between the two vectors.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what a great circle and a rhumb line are. Before any calculation, guess which of two routes is shorter and by how many percent.
- Choose the routesPick three to five city pairs at different latitudes and different east-west spans. Record each coordinate with its source.
- Build the vectorsDerive the unit vector for latitude and longitude, check it has length 1, and define every symbol.
- Trial run (1 lesson)Do one pair by hand and in a spreadsheet, and compare with a published distance. Fix any mismatch now: radians, signs of west and south.
- Get the distanceFind the angle from the scalar product and the distance from arc length. Repeat with a second method and agree to a stated precision.
- Add the constant-bearing routeDefine a flat map and a straight line on it, work out the length and compare it with the great circle for each pair.
- Go further, with a reasonPick one extension: the highest latitude of a route, the haversine form, or a flattened Earth. Say why it adds to your question.
- Test against real flightsCompare your distances with published flight distances. Explain the differences with numbers.
- Reflect as you goAfter every result, write one line on what it means for a real flight and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Coordinates to unit vectors – Write and check for each city.
- Scalar product – Find the angle between two cities from , then distance (AA HL).
- Chord and cosine rule – SL route: find the straight-line chord through the Earth with 3D distance, then the angle with the cosine rule in the triangle with the centre.
- Radians and arc length – Convert degrees to radians and use ; check units against a published distance in kilometres.
- Map-straight route – Define a flat map of latitude and longitude, work out the length of a constant-bearing route and write its excess over the great circle as a percentage.
Key equations
Syllabus links
- 3.13D distance, volume and surface area
- 3.2Right-angled trigonometry, sine and cosine rules
- 3.4Radians, arc length and sector area
- 3.5Unit circle definitions of sine, cosine and tangent
- 3.12Vectors: position, displacement, componentsHL
- 3.13Scalar product and angles between vectorsHL
- 3.13D distance, volume and surface area
- 3.2Right-angled trigonometry, sine and cosine rules
- 3.4Arc length and sector area
- 3.7RadiansHL
- 3.10Vectors: components, magnitude, positionHL
- 3.13Scalar and vector productsHL
Read up first
Data and sources to start from
Airport coordinates from an open airport database and published flight distances or tracked routes for several city pairs.
References to cite
- Snyder, J. P. (1987). Map projections: A working manual (U.S. Geological Survey Professional Paper 1395). U.S. Government Printing Office.
- Sinnott, R. W. (1984). Virtues of the haversine. Sky & Telescope, 68(2), 159.
- Williams, E. (n.d.). Aviation formulary. https://edwilliams.org/avform147.htm
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
radians, not degrees- Mixing degrees and radians in the arc length, so a distance is out by a factor of 57.
- Forgetting that west longitudes and south latitudes are negative, and getting a route on the wrong side of the Earth.
- Using the flat-map distance between two points as if it were a distance on the ground.
- Quoting an online calculator's output without deriving the formula behind it.
- Comparing with one real flight and calling the model wrong, when winds and airspace explain the gap.
- Treating the Earth as a perfect sphere without saying what error that causes.
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.