Apertia · Library
How can you compare two focal lengths at matched subject framing?
Match framing by doubling focal length and distance: 50 mm at 2 m and 100 mm at 4 m both cover 1367.052 mm, with near-equal depth but twice the background blur.
What matched framing keeps and changes
In the thin-lens model, the width of a flat subject that fills the frame is the sensor width divided by the magnification v ÷ u. Doubling both the focal length and the sensor-to-subject distance keeps that magnification, so 50 mm at 2000 mm and 100 mm at 4000 mm cover the same 1367.052 mm on a 36 mm sensor.
With framing matched, Apertia's depth-of-field formula gives almost the same total interval for both setups at the same aperture. The background-blur limit, f² ÷ (N × (u − f)), does not stay the same: it roughly doubles with the longer lens. That is the measurable part of the familiar claim that a longer lens blurs the background more at equal framing.
Worked example
The first thin-lens case uses a 50 mm focal length and a 2000 mm sensor-to-subject distance.
- Estimated width: 50 mm at 2000 mm (mm)
- 1367.051975
Worked example
Doubling both focal length and the total model distance gives the same flat-subject coverage in this ideal comparison. It does not establish an identical perspective or background appearance.
- Estimated width: 100 mm at 4000 mm (mm)
- 1367.051975
Compare two lenses at equal framing
- Fix the sensor preset and the flat width the subject should fill.
- Set lens A and its distance; read the scene width.
- Multiply the focal length and the distance by the same factor for lens B.
- Check that both scene widths match before comparing depth of field.
- Compare the background-blur limit with the background set to infinity in both cases.
Worked example at f/4
50 mm at 2000 mm, f/4, full frame: near 1837.237 mm, far 2195.405 mm, total 358.168 mm; background limit 0.329 mm. 100 mm at 4000 mm, f/4: near 3830.143 mm, far 4186.076 mm, total 355.933 mm; background limit 0.658 mm. The depth differs by about 2.2 mm, while the maximum background blur doubles from 0.329 to 0.658 mm.
- 50 mm · 2000 mm · total depth
- 358.168 mm
- 100 mm · 4000 mm · total depth
- 355.933 mm
- Background limit 50 mm / 100 mm
- 0.329 / 0.658 mm
Common mistakes
- Comparing two lenses from the same position and calling the result a matched-framing test.
- Matching the frame diagonally on sensors with different aspect ratios and assuming the height also matches.
- Reading equal depth of field as equal background appearance.
What the ideal comparison leaves out
Moving the camera changes perspective: background objects take up a different share of the frame, which no flat-subject calculation shows. Focus breathing can also alter real framing. The DOFMaster equations and Nikon's angle-of-view guide linked below cover the two parts of this comparison.