Science calculator

Thin-lens imaging.

Solve one conjugate distance from the other two, or enter focal length and a desired signed magnification to find both object and image positions.

Relationship

Focal length f

Object distance u

Image distance v

Image distance vSolved

—

signed thin-lens result

Object distance u: —
Image distance v: —
Transverse magnification m = −v/u: —
Image orientation: —

Display precision

\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

m = - \frac{v}{u}

u = \frac{f (m - 1)}{m}, v = f (1 - m)

f focal length · u object distance · v image distance · m transverse magnification

Assumptions

Thin lens in the paraxial approximation; distances are measured from the principal plane.
Sign convention: real object u > 0, real image v > 0 and converging lens f > 0. Virtual quantities are negative.
Magnification m < 0 means an inverted image; m > 0 means upright.
Aberrations, diffraction, lens thickness and aperture clipping are neglected.

How it works

The thin-lens equation links focal length with object and image distances. In the three-distance mode, leave any one blank and the calculator rearranges the reciprocal relation.

Combining the lens equation with m = −v/u means focal length and signed magnification determine both distances: u = f(m − 1)/m and v = f(1 − m). Negative m gives an inverted real image under this convention.

A converging lens with a real object beyond its focal point gives a positive real-image distance. Positive magnification describes an upright image; a virtual image has negative v. The degenerate values m = 0 and m = +1 do not give finite non-zero conjugates here.

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