Which function best describes the arch of a famous bridge?

AA SLAI SL Accessible Also in: Geometry & Voronoi, Art & music

Research question

Is the arch of [a chosen bridge] better modelled by a quadratic, a cosine curve or a semi-ellipse, judged by the sum of squared residuals?

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

Why it makes a good exploration

Arches are chosen for strength, not looks, so the “right” curve depends on how the bridge carries its load. Comparing three families of curves on a structure you have visited gives a clear, personal modelling question with a satisfying answer.

The mathematics you'll need

Quadratic, sinusoidal and elliptical models
Transformations of functions to fit a model through known points
Residuals and sum of squared residuals
Symmetry and domain restrictions

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

Use a front-on photograph with a known dimension (span or height from the bridge's published details) and read 15+ points in GeoGebra.

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

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

Choose a bridge you have a connection to; find its span and rise.
Set axes and extract points from the photograph.
Fit each family through the key points, then improve with least squares.
Compare residuals; discuss which shape the engineers probably intended and why.

Pitfalls that cost marks

Photos taken off-centre make the arch asymmetric.
Fitting only through three points and calling it a model — use all your data to judge fit.
Forgetting to convert pixels to metres consistently.

Showing personal engagement

Visit and photograph the bridge yourself.
Look up how the bridge was designed and test whether your best-fitting curve agrees.
Compare two bridges of different eras or materials.

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

Taking it further

Extend by computing the area under the arch (for the opening) by integration or the trapezoidal rule, and compare the three models.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).