Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How are letters drawn on a screen? Bézier curves

AA HLAA SLAI HL Solid Also in: Pure maths, Calculus

Research question

How do cubic Bézier curves define the outline of a letter, and how many control points are needed to approximate a circle to within 0.1% of its radius?

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Why it makes a good exploration

Every font and logo on your screen is drawn with Bézier curves. Deriving them, reconstructing a letter, and finding the famous “magic number” for approximating circles is creative and precise.

The mathematics you'll need

Parametric equations and vectors
Bernstein polynomials (explain)
Derivatives of parametric curves (tangents)
Error of approximation and optimisation

Where the data comes from

Trace a letter from a font in GeoGebra or Desmos.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

A possible outline

Derive linear, quadratic and cubic Bézier curves.
Show the tangent property at endpoints.
Reconstruct a letter.
Approximate a quarter circle and find the optimal control distance.
Reflect on smoothness where curves join.

Pitfalls that cost marks

Using software output without the equations.
Tangent continuity ignored at joins.
Error measured only at a single point.

Showing personal engagement

Design your own initial as a logo.
Compare a serif and a sans-serif font.
Measure how the circle error changes with the control distance.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Parametric equations and vectors; Bernstein polynomials (explain)
AA HL	Good fit	Parametric equations and vectors; Bernstein polynomials (explain)
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	Parametric equations and vectors; Bernstein polynomials (explain)

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: using software output without the equations; and tangent continuity ignored at joins. Say how each one affects your answer.

Use of mathematics (E)

SL: parametric equations and vectors; bernstein polynomials (explain); derivatives of parametric curves (tangents); error of approximation and optimisation, each explained in your own words and carried out correctly.

How art & music explorations reach the top bands in general: Art, music & design IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Derive the value k ≈ 0.5523 for the circle approximation and explain the optimisation.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL art & music exploration adds.

See a complete IA, marked

Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete art & music exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA HL, AA SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Parametric equations and vectors, bernstein polynomials (explain), derivatives of parametric curves (tangents) and error of approximation and optimisation.

Where can I get data for this IA?

Trace a letter from a font in GeoGebra or Desmos. Checked sources and tools: Desmos graphing calculator and GeoGebra.

Turn this idea into your IA

The planner opens with “How are letters drawn on a screen? Bézier curves” filled in: the question, the maths and the data sources.

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