IA idea · Matrices, transformations & Markov chains

How does a game draw 3D on a flat screen? Projection matrices

AI HLAA HL Ambitious Also in: Trig & navigation, Art & music

Research question

How do matrices rotate a 3D model and project it onto a 2D screen, how does the camera's distance change the perspective, and how well does your projection match a photograph of the real object?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

It explains how every 3D game works, and comparing with a photo you take gives a real test of the model.

The mathematics you'll need

3D rotation matrices
Perspective projection by similar triangles
Homogeneous coordinates (new)
Parallel versus perspective projection
Comparing predicted and measured image positions

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Measure a simple real object (a box, a building) and photograph it from a known position.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Set up coordinates for the object.
Derive the projection with similar triangles.
Write it as a matrix and add rotations.
Compare with your photograph.
Reflect on lens distortion and measurement error.

Pitfalls that cost marks

Copying a graphics library's matrices without explanation.
Not checking with a known simple case.
Ignoring lens distortion in the comparison.

Showing personal engagement

Model your own room or school building.
Recreate a photo you took.
Find the camera distance that best matches the photo.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	3D rotation matrices; Perspective projection by similar triangles
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	3D rotation matrices; Perspective projection by similar triangles

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Collect the data for your matrix yourself (counting transitions, measuring a shape), and choose the states or the transformation from a situation you care about.

Reflection (D)

Question the model's assumptions: is the process memoryless, are the probabilities constant, does the transformation preserve what it should? Say how each affects your conclusion. For this idea, start with: copying a graphics library's matrices without explanation — say how it affects your answer.

Use of mathematics (E)

SL: Matrices are in AI HL. At SL, keep to small matrices you explain carefully as new mathematics, with every multiplication shown once and the result interpreted.

HL: Transition matrices, powers, steady states and eigenvalues used correctly, with diagonalisation or a general result derived, and the long-run behaviour interpreted in context.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Find the camera position from a photo (the inverse problem).

Extending it for HL

Diagonalise the matrix to find a formula for the nth state, or compare the steady state with what your data actually shows.

See a complete IA, marked

Our annotated exemplar How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) asks a different question, but shows how a complete matrices & markov exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).