Maths IA · Geometry and trigonometry · topic 62 of 97
How does a phone find itself? Trilateration · IB Maths IA idea: guide, sample graph and step-by-step plan
distances in, position outYour phone doesn't know where it is. It only knows how long signals took to arrive from points it can't see. From those times it finds a spot on the Earth, and the geometry of the beacons decides whether that spot is good.
- ✓ you like problems where a few measurements pin down an unknown position
- ✓ you are happy to set up a system of equations and solve it with a matrix
- ✓ you want to know why a phone is sometimes right to 3 metres and sometimes wrong by 30
The big idea
A distance to one known point puts you on a circle (in 3D, a sphere). Distances to three points put you at the meeting point of three circles. Each distance gives a quadratic equation, but subtracting one equation from another cancels the squares and leaves a linear one, so the problem becomes a matrix equation. Real receivers have a clock far worse than the satellites', so there is a fourth unknown: the clock error. Then ask how errors in the distances spread into errors in the position. The answer depends on where the beacons are, not only on how accurate they are. The set-up is routine. The marks come from the error analysis: derive how geometry magnifies range error, simulate it, and compare with a real dataset.
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 →
Use three beacons and circles in the plane. Solve with equations of circles, then again by subtracting equations to get lines.
→ Two methods, one answer. Show they agree and say which one extends to 3D.Work in 3D with four satellites and an unknown clock offset, and solve for four unknowns.
→ A 4 by 4 system. Show that a clock error of one microsecond moves the fix by hundreds of metres if it is not solved for.Place the beacons close together, then spread them out, and measure how the position error changes for the same range error.
→ A formula for the error multiplier in terms of the angle between lines of sight, checked by simulation.Use five or six beacons with noisy distances and find the best-fit position.
→ A least-squares fit. Compare it with solving using the best four beacons only.Use real satellite positions and receiver data from a public source and compute your own fix.
→ Your model against a published position. Every gap (atmosphere, clock, rounding) becomes evaluation.Let the beacons move along known paths and track how the fix changes through a day.
→ Position as a function of time, and an answer to when the geometry is best.Your Maths IA research question: run the check
2D or 3D? what error?Write your own question first. Then make sure it ticks every box:
- Names the set-up: 2D or 3D, how many beacons, and where their coordinates come from.
- Names the quantity you will find: a position error, a multiplier from geometry, or the effect of a clock error.
- Says how the result is checked: a known position, a simulation, or published receiver data.
- Says what is assumed about the distance errors (size and independence) and why.
- Goes beyond solving a system once: there is an error analysis or a comparison of designs.
The model
Three beacons at known places, each giving a distance to you. The circles meet at your position, and subtracting two circle equations leaves a straight line. Draw yours with real coordinates.
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 this problem, how a receiver gets distances from signal times, and what you expect before calculating.
The distance equations for each beacon in 2D, then 3D, with every symbol defined.
Subtracting equations, the matrix form and the solution. A worked example checked by a second method.
Adding the fourth unknown, solving the 4 by 4 system and showing what an uncorrected clock does.
How range errors become position errors, with a formula, a simulation and one graph.
Your fix beside a published position. A number for the difference and a reason for it.
Back to your aim, with numbers: which geometry is best, and by how much.
Flat Earth against a real one, ignored atmosphere, the size of the noise you assumed. Say which assumption matters most.
Where every coordinate, signal fact and dataset came from.
What your Maths IA graph could look like
state the noise you addedA made-up simulation to show the form of a strong graph, not the answer: how much a range error is multiplied into a position error, against the angle between two beacons. Yours will look different. example only
- A theory curve and simulated points on one set of axes, so agreement is visible.
- Both axes are ratios, so say clearly that they have no units.
- State how many trials sit behind each point, and fix the random seed.
- Mark the geometry of a real receiver on the curve, then explain where it lands.
- Another graph to consider: position error against time as the satellites move across the sky.
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 how a receiver turns signal times into distances. Before any calculation, guess which beacon layout gives the smallest error.
- Choose the set-upPick 2D or 3D, the number of beacons and their coordinates: invented but sensible, or real satellite positions.
- Write the equationsWrite one distance equation per beacon. Define every symbol and say what is known and what is unknown.
- Trial run (1 lesson)Choose a point, work out its distances to three beacons, then recover the point from those distances. If it fails, fix it now.
- Make it linearSubtract equations, write the matrix form and solve. Check the answer by substituting back into the original equations.
- Add noise and the clockAdd random errors to the distances and a clock offset, and show how the fix moves. Fit the extra unknown.
- Go further, with a reasonDerive how geometry multiplies range error, or fit the best position to extra measurements. Say why it answers your question.
- Test with real dataCompare your method with real satellite or receiver data from a public source, and put a number on any gap.
- Reflect as you goAfter every result, write one line on what it means for a real receiver and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Equations of circles and spheres – Write for each beacon , in 2D and in 3D.
- Subtracting to a linear system – Subtract the first equation from the others so the quadratic terms cancel, leaving to solve with matrices.
- Clock error as a fourth unknown – Add to every distance and solve four equations for position and (AA HL, AI HL).
- Error propagation – For two lines of sight at angle , show the position error is about for range error .
- Simulation – Add random errors to the distances, solve many times and compare the spread of the results with the formula.
Key equations
Syllabus links
- 1.16Systems of up to three linear equationsHL
- 2.10Solving equations graphically and analytically
- 3.13D distance, volume and surface area
- 3.12Vectors: position, displacement, componentsHL
- 3.13Scalar product and angles between vectorsHL
- 1.8Solving systems and polynomial equations with technology
- 1.14Matrices, determinants and inversesHL
- 3.13D distance, volume and surface area
- 3.5Perpendicular bisectors
- 3.10Vectors: components, magnitude, positionHL
- 4.3Averages and spread
Read up first
Data and sources to start from
Invented beacon layouts with simulated noise, plus (for the best marks) real satellite positions or receiver data from a public source.
- GPS.gov: official U.S. government GPS information
- International GNSS Service (IGS): public satellite orbit and clock products
References to cite
- Strang, G., & Borre, K. (1997). Linear algebra, geodesy, and GPS. Wellesley-Cambridge Press.
- Kaplan, E. D., & Hegarty, C. J. (Eds.). (2017). Understanding GPS/GNSS: Principles and applications (3rd ed.). Artech House.
- GPS.gov. (n.d.). Official U.S. government information about the Global Positioning System. https://www.gps.gov/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
geometry matters as much as accuracy- Solving the quadratic equations directly and getting two answers, without saying how you pick the right one.
- Leaving out the clock error and then wondering why the fix is hundreds of metres off.
- Using beacons that are nearly in a line from the receiver and trusting the result.
- Adding noise to the position when it belongs on the distances.
- Mixing units, so that metres, kilometres and seconds all appear in one equation.
- Copying a satellite position table without saying which frame and which time it is for.
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.