Signed Mirror Equation and Image Magnification

Formula and method

For a concave mirror with an object at u=−30 cm and focal length f=−10 cm, the image is at v=−15 cm. Signed magnification is −0.5 and areal magnification is 0.25.

1/f=1/u+1/v; r=2f; m=−v/u; A=(v/u)². For a plane mirror, v=−u, m=A=1 and focal length/radius are unbounded.

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• The source uses Cartesian signed distances, not the convention in which real object distance and concave focal length are positive. Keep one convention throughout: real-object u<0, concave f<0, convex f>0; image v<0 is real and v>0 virtual. Signed magnification m=−v/u indicates orientation, while areal magnification A=m² discards orientation.
• The spherical-mirror relations use the paraxial approximation and r=2f. Large apertures, off-axis rays and spherical aberration are outside this scalar model.
• A finite convex mirror with a real object gives 0<m<1 and 0<A<1. A concave mirror gives negative m for a real inverted image, or m>1 for an upright virtual image. A plane mirror gives m=A=1, v=−u and has unbounded focal length and radius; the two original forced infinity outputs are represented by this explicit note.
• Areal magnification alone cannot determine signed image orientation. The retained concave inverse v=u√A chooses the real-image negative-v branch. Supply signed linear magnification or focal length to identify a virtual-image branch. Given signed v and A, u=−|v|/√A is the unique real-object distance. No claim of a unique image branch from A alone is made.
• An object at the focal position has no finite image distance. That singular result remains an error; the plane-mirror presentation correction does not hide nonfinite results in other controls.
• Controls use bounded exact finite-decimal preflight. Nonzero values below 10⁻¹⁰⁰⁰⁰ in equation base units are unsupported. Arithmetic uses 64 significant digits; tiny finite results use scientific notation and an overflowing or zero-losing unit conversion is withheld with an error. Near the focal singularity, rounding can limit the reliable image distance.

Common questions

References

• OpenStax: spherical mirror equation and magnification
• OpenStax: images formed by plane mirrors
• Calculation definition and unit reference

Correct unique physical signs in convex/plane areal inverses for u<0,v>0, and concave object inverse u=-abs(v)/sqrt(A) for both real and virtual images. Preserve the captured concave image-from-area negative-v branch and all signed linear magnification/focal length inverse controls. All13originalequations/variants are pinned. Preserve all17originalcondition bodies; replace sign comparisons equivalently and append physical image/magnification domains. Plane f/r are FORCED output-only and never finite in the original graph: only these exact two display fields become hidden, while the authored sign-convention note preserves the unbounded curvature/focal-length meaning. No generic nonfinite suppression.

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