Engineering
Thin-Lens Equation (1/f = 1/do + 1/di), Magnification & Diopter formulas
Solve image distance di = (f * do) / (do - f), transverse magnification M = -di / do, optical power (Diopters), and Real vs Virtual image type.
Machine-vision & optical bench calculations: Real inverted vs Virtual upright images
Gaussian Thin-Lens optics relates Focal Length f (A2, positive for converging convex lenses, negative for diverging concave lenses), Object Distance do (B2), and Image Distance di via 1/f = 1/do + 1/di -> solved directly as di = (f × do) / (do - f) (guarded when do == f, where rays exit parallel at infinity)!
Placing an object at do = 150 mm in front of an f = 50 mm machine-vision lens forms a sharp REAL image at di = (50 × 150) / (150 - 50) = +75.00 mm behind the lens, with Transverse Magnification M = -di / do = -75 / 150 = -0.50 (half size, negative = inverted!). Move the object inside the focal point to do = 30 mm (< 50 mm), and di flips to -75.00 mm (negative = VIRTUAL, upright magnifier M = +2.50×)! Optical Power in Diopters = 1000 / f_mm (+20.0 D for f = 50 mm).
Use the thin-lens, paraxial approximation with a real object: object distance is positive, focal length is nonzero and may be negative for a diverging lens. Real multi-element camera lenses and off-axis aberrations require more detailed modeling. Values very close to the focal plane produce very large distances and are sensitive to input precision.
| Lens Setup (f = +50 mm = +20 Diopters) | Image Distance di (mm) | Magnification M = -di/do | Image Classification |
|---|---|---|---|
| Object do = 150 mm (> 2f) | +75.00 mm | -0.50× (Reduced, Inverted) | Real Image (di > 0) |
| Object do = 100 mm (= 2f, 1:1 macro) | +100.00 mm | -1.00× (1:1 Life-Size) | Real Image |
| Object do = 30 mm (< f, loupe) | -75.00 mm | +2.50× (Magnified, Upright!) | Virtual Image (di < 0) |
Image Distance di (mm) and Transverse Magnification M formulas
Guards B2 = A2 ("Infinity (Collimated)") when an object sits right at the focal plane.
=IF(OR(COUNT(A2:B2)<>2,A2=0,B2<=0),"Invalid inputs",IF(B2=A2,"Infinity",ROUND(A2*B2/(B2-A2),2)))Transverse Linear Magnification M = -di / do
Negative M indicates an inverted real image on a camera sensor; positive M indicates an upright virtual image.
=IF(OR(COUNT(A2:B2)<>2,A2=0,B2<=0),"Invalid inputs",IF(B2=A2,"No finite image",ROUND(-A2/(B2-A2),4)))Try the example
Use the original values or change them to test your understanding.