Apertia · Library
Object distance and image distance in close-up photography
Audit the distance split with u² − d·u + d·f = 0: for d = 1000 mm and f = 100 mm the roots are u = 887.298 mm and v = 112.702 mm, and u × v = d × f.
One quadratic gives both distances
Combining d = u + v with the thin-lens equation 1/u + 1/v = 1/f gives u² − d·u + d·f = 0. Its two roots are u and v themselves: they add up to d and multiply to d × f. Apertia takes the larger root, u = (d + √(d² − 4df)) ÷ 2, as the object distance and uses the smaller one as the image distance, so the subject is always on the far side of the one-to-one point.
This gives three checks you can run on any row of the table above: u + v must return d, u × v must equal d × f, and 1/u + 1/v must equal 1/f. If one fails while the others pass, the most likely cause is a rounded value copied from a display rather than the stored number.
Three finite close-focus examples
The focal length stays at 100 mm. The table retains both parts rather than rounding the image distance to the focal length. Values are displayed to six decimal places; the checks use the unrounded calculation.
| Sensor-to-subject d (mm) | Object u (mm) | Image v (mm) | u + v − d (mm) | Magnification v/u |
|---|---|---|---|---|
| 400 | 200 | 200 | 0 | 1 |
| 1000 | 887.298335 | 112.701665 | 0 | 0.127017 |
| 3000 | 2896.424004 | 103.575996 | 0 | 0.03576 |
Audit a row by hand
- Take d and f from the row and compute the discriminant d² − 4df.
- Take its square root and form (d + root) ÷ 2 and (d − root) ÷ 2.
- Check that these match u and v in the table.
- Multiply u by v and compare with d × f.
- Add 1/u and 1/v and compare with 1/f, using unrounded values.
Worked example: d = 1000 mm, f = 100 mm
The discriminant is 1000000 − 400000 = 600000, whose square root is 774.596669. The roots are (1000 + 774.596669) ÷ 2 = 887.298335 mm and (1000 − 774.596669) ÷ 2 = 112.701665 mm. Their product is 100000, which is d × f, and 1/887.298335 + 1/112.701665 = 0.01 = 1/100. For the d = 3000 mm row the discriminant is 7800000 and its root 2792.848009, giving the 2896.424004 and 103.575996 mm in the table.
- d² − 4df
- 600000
- √(d² − 4df)
- 774.596669
- u × v
- 100000
- 1/u + 1/v
- 0.01 mm⁻¹
Common mistakes
- Taking the smaller root as the object distance, which describes the mirrored arrangement with the subject close to the lens.
- Checking 1/u + 1/v with six-decimal display values and treating the tiny residual as an error in the model.
- Using a d below 4f, where the discriminant is negative and no real root exists.
What the audit covers
These checks show that the model's numbers are internally consistent. They do not locate the principal planes of a real lens or calibrate its focus scale. The depth-of-field calculator linked below applies the same split to every distance you enter.