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Maths IA · Calculus · topic 28 of 97

Which aperture gives the sharpest photo? · IB Maths IA idea: guide, sample graph and step-by-step plan

two blurs, one best stop

Open the aperture wide and the lens blurs the picture. Close it down and diffraction blurs it instead. So there is a best f-number in the middle, and you can find it with calculus and then check it with a camera.

Difficulty●●○ Medium
KitA camera with manual aperture, a tripod, a test chart or a printed edge, free image-analysis software, a spreadsheet
TimeAbout 12 to 16 hours
FitsAA SL3 of 3AA HL3 of 3AI SL2 of 3AI HL3 of 3
  • ✓ you take photos and want to know why one setting looks sharper
  • ✓ you like a model with two competing effects and a best choice in between
  • ✓ you want a real experiment with a graph you made yourself

The big idea

The f-numbers on a lens form a geometric sequence, each stop about 1.4 times the last. Two kinds of blur compete as you change the aperture. One comes from the lens and shrinks as you close the aperture; the other comes from diffraction and grows. Add them in quadrature and you have a function of the f-number with one minimum, which you can find with the derivative. The minimum depends on only two constants, and you can fit both from your own test shots. Then ask how wide the sweet spot is: the range of apertures within 95% of the best sharpness is a quadratic inequality you can solve and test. A good IA asks whether the model survives a different lens.

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.

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Fit the two constants

Photograph a test chart or sharp edge at every f-stop, measure a sharpness number for each, and fit the two constants of the model.

→ You turn measurements into parameters, and your fit gives a best aperture that you can compare with the one the photos suggest.
The sweet-spot range

Find the range of f-numbers that keep at least 95% of the best sharpness, by solving an inequality.

→ A quadratic inequality in the square of the f-number. You can say how many real stops fall inside the range.
Change the lens

Repeat the measurements with a second lens or a different focal length.

→ The constants change, and so does the best aperture. You decide whether one model can describe both.
A different aberration law

Replace the lens blur term with another power of the f-number, and fit the exponent as well.

→ The minimum can no longer be written in a simple closed form. You solve numerically and say whether the extra parameter helps.
Low light

Add the shutter speed. Closing the aperture needs a longer exposure, and a longer exposure adds shake blur.

→ A third blur and a constraint between aperture and time. You optimise under the constraint, at HL with a second variable.
Corner against centre

Measure sharpness at the centre and at the corner of the frame, and fit each separately.

→ Two best apertures. You compare them and discuss which matters for a given photo.

Your Maths IA research question: run the check

which lens? how is sharpness measured?

Write your own question first. Then make sure it ticks every box:

  • Names the camera and lens, the focal length and the test subject, and how sharpness will be measured.
  • Says what is being fitted (the two constants) and what is being predicted (the best f-number and the range).
  • Says how the model will be compared with the measurements, with a number for the agreement.
  • States what the model leaves out, such as focus errors, camera shake, sensor resolution and lens copy variation.
  • Is narrow enough for 12 to 20 pages and includes a decision of your own, such as how to combine the two blurs.
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The model

Two apertures, one lens. Wide open, the cone of light is wide and the lens does badly. Stopped down, the cone is thin and diffraction spreads the spot. Draw your own lens and mark what you measure.

sensorapertureDfocal length fN = f / Dwide open: small Nwide conestopped down: large Nnarrow conelens blur ~ A / N (model)shrinks as N growsdiffraction blur= 2.44 λ Ngrows as N grows

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.

Aim and context~1 page
Why this lens, what 'sharp' means, and what you expect before calculating: wide open, stopped down, or somewhere between?
The f-number sequence~1.5 pages
Why the stops form a geometric sequence, and the f-numbers as a formula in the stop number.
Two blurs, one model~2 pages
Each blur as a function of the f-number, every symbol defined, and the total blur and the sharpness.
The best aperture~2 pages
The minimum found with the derivative, and the 95% range found by solving an inequality.
Fitting real photographs~3 pages
Your test shots, how sharpness was measured, the fitted constants and the fit against the data.
How good is the model?~1.5 pages
Predicted and measured best aperture side by side, with uncertainty, and a reason for each gap.
Conclusion~1 page
Back to your aim: the best aperture, how wide the sweet spot is, and how sure you are.
Evaluation~2 pages, the biggest part!
Which effects are missing, how far one lens can speak for others, and which assumption matters most.
Bibliographynot counted
Where every formula, constant and measuring method came from.

What your Maths IA graph could look like

show the 95% range!

Made-up numbers to show the form of a strong graph, not the answer: sharpness against f-number on a log axis, with a fitted curve and measured points. Yours will look different. example only

15913172120406080100f-number NRelative sharpness (% of best)model: 1 / total bluryour measured sharpness (made-up)the peak is broad: one stop eitherside loses only a littleILLUSTRATIVE · MADE-UP DATA
  • Sharpness against f-number, with the fitted model and your own measured points on one set of axes.
  • Axis labels give units: the f-number has none, sharpness is a percentage of the best.
  • Mark the predicted best aperture and the range within 95%, then compare with the points.
  • Error bars from repeated shots show whether the difference between stops is real.
  • Another graph to consider: the two blur terms and their total on a log axis, to see why the minimum appears.

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.

Two panels. Left: log-log plot with a falling dashed line, a rising dashed line and a solid U-shaped total with its minimum marked at f/5.6. Right: a bell-shaped sharpness curve against f-number with dots at each whole stop and the region above 95 percent shaded.
(a) Blur from the lens (A/N), blur from diffraction (2.44 times wavelength times N) and their quadrature total on a log axis, with the minimum at f/5.6. (b) Relative sharpness: the range within 95% of the best runs from f/4.4 to f/7.1.Source: Computed from the model with an assumed lens constant A (chosen so the best aperture is f/5.6) and wavelength 550 nm; not measured data.

How much sharpness does each stop give?

Rank seven stops on one number: sharpness as a percentage of the best. Work it out for your own lens and explain why the loss is not the same either side of the best stop.

f/250%f/2.869%f/490%f/5.6100%f/889%f/1170%f/1649%one stop from the best costs about a tenth of the sharpnessRelative sharpness from the model, best aperture f/5.6 (assumed lens constants)ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictFind out what the f-number means and what diffraction does. Before any calculation, guess which aperture on your lens gives the sharpest picture.
  2. Choose the lens and subjectPick one lens, one focal length and a flat, well-lit test target. Fix everything except the aperture, and say how you did that.
  3. Show the stops are a sequenceWrite the f-numbers as a geometric sequence and explain the ratio from the area of the opening.
  4. Trial run (1 lesson)Shoot the target at three apertures and measure sharpness. Is the measurement repeatable? Fix your method now.
  5. Build the modelWrite each blur as a function of the f-number, combine them in quadrature, and define sharpness. Graph it with made-up constants first.
  6. Find the minimumDifferentiate the total blur or its square, find the best f-number in terms of the two constants and show it is a minimum.
  7. Fit and compareFit the two constants to your measurements. Compare the predicted best aperture and the 95% range with the photos.
  8. Go further, with a reasonChange the lens, fit an exponent or add shutter speed. Say what question the extra step answers. HL: this is where the rigour for criterion E comes from.
  9. Reflect as you goAfter every result, write one line on what it means for a photographer and what it assumes. These lines become your evaluation.
  10. Cut, check, then sendCheck units everywhere, every graph labelled, every symbol defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

  • Geometric sequence – Write the f-numbers as and explain the ratio from halving the area of the opening each stop.
  • Modelling with two effects – Combine blur terms and in quadrature to get the total blur and the sharpness .
  • Differentiation and optimisation – Minimise to get and show it is a minimum.
  • Quadratic inequality – Solve as a quadratic in for the range of apertures within 95% of the best.
  • Fitting and residuals – Fit and to measured sharpness, compare predicted and measured best aperture and report the uncertainty.
  • Log scales – Plot against or on a log axis so that equal stops are equally spaced.

Key equations

Syllabus links

  • 1.3Geometric sequences and series
  • 2.7Quadratic equations, inequalities and the discriminant
  • 2.9Exponential and logarithmic functions
  • 5.6Standard derivatives; chain, product and quotient rules
  • 5.8Maxima, minima, inflexion and optimisation

Read up first

Data and sources to start from

Photograph a test chart or a sharp edge at every f-stop of one lens on a tripod, and measure a sharpness number (for example the spatial frequency where contrast falls to 50%) with free image-analysis software.

References to cite

  1. Hecht, E. (2017). Optics (5th ed.). Pearson.
  2. Ray, S. F. (2002). Applied photographic optics (3rd ed.). Focal Press.

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

stops are geometric
  • Adding the two blurs directly when you have said you add them in quadrature, or the other way round, without saying why.
  • Treating the f-numbers as an arithmetic sequence. They are a geometric one, so plot them on a log axis.
  • Measuring sharpness with a different focus or light at each aperture, so the aperture is not the only thing that changed.
  • Using one shot at each f-stop with no repeats, then reading meaning into differences smaller than the shot-to-shot noise.
  • Reporting the best aperture to three decimals when the real stops are only whole steps.
  • Assuming your fitted constants apply to every lens, or to the corner of the frame.

More Maths IA ideas like this

Make sure it's yours

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.

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