Maths IA · Geometry and trigonometry · topic 61 of 97
Fonts and logos: how do Bézier curves work? · IB Maths IA idea: guide, sample graph and step-by-step plan
four points, one curveThe letters on this page are made of curves, and each curve is stored as just three or four points. You're going to find out how four points make a smooth curve.
- ✓ you like design, lettering or logos
- ✓ you enjoy building a curve from a handful of numbers
- ✓ you want a topic where polynomials, vectors and error all meet
The big idea
A cubic Bézier curve is built from four control points. The curve starts at the first, ends at the last, and is only pulled towards the two in the middle. Mix the points step by step and you get de Casteljau's construction; multiply it out and each coordinate is a cubic polynomial in t whose weights come from the binomial theorem. The control points are the whole design: the first and last tangents point along the control polygon, and two curves join smoothly when their tangents line up. The test of the model is a shape you know exactly. How well can a few cubics draw a circle? Measure the error, choose the control points to shrink it, and say what 'good enough' means for a screen or for a printer.
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 →
Build a point with three rounds of mixing, then expand the result as a cubic in t with binomial weights. Show both agree.
→ Two routes to one formula, each checked. The binomial link is a proof, not a lookup.Place the middle control points so a cubic is a quarter circle. Measure how far it strays from the true circle.
→ An optimisation with an error you can state as a percentage. Compare two choices of control distance.Join two cubics so the join has no corner, then no jump in curvature, and say which one the eye can see.
→ Derivatives of a parametric curve decide the design. HL: justify each condition.Find the arc length of a curve numerically and compare it with a chord-sum estimate.
→ A numerical-integration question: how many pieces are enough for a stated accuracy?Fit control points to a letter or simple logo shape you drew yourself, then measure the gap between curve and outline.
→ A fitting problem with your own data and a clear error measure.Replace a cubic by a quadratic one (three points) and compare what shapes each can make.
→ A change in degree, and a reason fonts use both kinds.Your Maths IA research question: run the check
which shape? which error?Write your own question first. Then make sure it ticks every box:
- Names the curve type (quadratic or cubic) and the shape you will draw or test it on.
- Says what you will measure: tangent direction, arc length or the distance from an exact shape.
- Says how you will judge success: an error in percent of the size, with the points you sample at.
- Explains which results you derive yourself and which come from software.
- States what you leave out: stroke width, curvature continuity or three-dimensional curves.
The model
Four control points and the control polygon between them. Draw your own curve and mark the point you build for one value of t.
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.
Where Bézier curves are used, which shape you will draw and what you expect before calculating.
Mixing the points step by step, then the cubic in t with every symbol defined.
Why the weights are binomial, proved for the cubic, and what they sum to.
Derivatives at the ends, the join condition and one worked join with numbers.
The control distance, the distance from the circle as a function of angle and the largest error.
Arc length by numerical integration, or a letter fitted to control points, with a measured gap.
Back to your aim: how few points are enough, with the error as a number.
Errors you measured at a few points only, rounding in software, what a real font editor also needs. Say which limit matters most.
Where every formula, tool and shape came from.
What your Maths IA graph could look like
label the end values!A made-up curve to show the form of a strong graph, not the answer: each coordinate of a cubic Bézier curve against t, with points you read off a traced outline. Yours will look different. example only
- A coordinate against t shows the cubic itself. The picture in the plane is a second graph.
- Axis labels give units, even if the unit is only 'font units'.
- Mark the two end heights: they are the only control values the curve actually reaches.
- Plot points read from a real outline as dots. A gap between dots and curve is evidence, and a number for it belongs in your text.
- Another graph to consider: the distance from an exact circle against the angle.
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 what a control point does. Before calculating, guess where the curve goes if you drag P₁ far away.
- Pick the shapeChoose a circle, a letter or a simple logo you drew yourself, so you have an exact or measurable target.
- Build the curve by mixingDo the three rounds of mixing for one value of t on paper, then check the point in software.
- Trial run (1 lesson)Write x(t) and y(t) as cubics and plot them. Check the end points and the end tangents against your drawing before going on.
- Prove the weightsShow the four weights come from expanding (1 − t + t)³ and that they add to 1 for every t.
- Add tangents and a joinDifferentiate, find the direction at each end and build a smooth join between two curves.
- Test against the targetMeasure the gap between curve and target at many points. Report the largest gap as a percentage of the size.
- Go further, with a reasonOptimise the control distance to shrink the error, or find arc length numerically. HL: justify the method and its accuracy.
- Reflect as you goAfter every result, write one line on what it means for a designer and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every symbol is defined once, every graph has labelled axes and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Binomial expansion – Expand to get the four weights and show they sum to 1.
- Parametric differentiation – Differentiate and to find the tangent at each end, and , and the join condition.
- Vectors – Write each point as a weighted sum of the control point vectors; test collinearity of tangents at a join.
- Solving for a parameter – Require the mid-point to lie on the unit circle and solve for the control distance .
- Error and numerical methods – Measure the radial error at many angles; estimate arc length with the trapezoidal rule or Simpson's rule.
Key equations
Syllabus links
- 1.9Binomial theorem
- 2.6Quadratic functions
- 3.4Radians, arc length and sector area
- 5.3Differentiating powers of x
- 5.5Integration as anti-differentiation; areas
- 3.12Vectors: position, displacement, componentsHL
- 3.13Scalar product and angles between vectorsHL
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 3.10Vectors: components, magnitude, positionHL
- 5.3Differentiating powers of x
- 5.8Trapezoidal rule
Read up first
Data and sources to start from
Control points you choose, and points read from an outline you drew or an open-licence font glyph, compared with the curve at sample values of t.
References to cite
- Farin, G. (2002). Curves and surfaces for CAGD: A practical guide (5th ed.). Morgan Kaufmann.
- Mortenson, M. E. (2006). Geometric modeling (3rd ed.). Industrial Press.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
control points are not on the curve- Saying the curve passes through all four control points. Only the first and last are on it.
- Reading a curve's tangent from the picture instead of differentiating x(t) and y(t).
- Treating the dashed control polygon as part of the answer. It is the instructions, not the curve.
- Claiming a circle is 'exactly' drawn by cubics. State the error and how you measured it.
- Using a software arc-length figure with no check against an estimate of your own.
- Measuring the gap at the join points only, where the error is smallest.
More Maths IA ideas like this
Hanging chains: catenary or parabola?●●○
Bridge arch: is a cubic or quartic a real shape?●○○
Drawing a shape with functions: how close can you get?●○○
Roots of unity: a pentagon by ruler and compass?●●●
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.