IA idea · Art, music & design
How are letters drawn on a screen? Bézier curves
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?
Adapt it: change the place, the data or the comparison until the question is yours.
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
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
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.
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
- 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.
Taking it further
Derive the value k ≈ 0.5523 for the circle approximation and explain the optimisation.