Signed Height Audit - Optics Step by Step

A signed-height audit reads the image orientation off the magnification m = -d_i / d_o, with a negative sign flagging an inverted image and a positive sign an upright one. The arithmetic is exact only for the stated heights, distances, and sign conventions in the ideal thin-lens or mirror model. Real measurements add alignment, lens/mirror aberrations, finite aperture, thickness, screen/readout calibration, sign-convention bookkeeping, finite precision, and measurement uncertainty.

Signed magnification turns a positive object height into an inverted or upright image height. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Magnification is a signed height contract

Use object height 4 metres. The signed image height is object height multiplied by magnification.

h_{\text{image}} = m h_{\text{object}}

The same object height can draw different image heights

Hold object height fixed and change the object placement. The same height can produce a smaller inverted image, a taller inverted image, or an upright virtual image.

\begin{array}{c|c|c|c}u&v&m&h_{\text{image}}\\30\ \text{m}&15\ \text{m}&\tfrac{-1}{2}&-2\ \text{m}\\15\ \text{m}&30\ \text{m}&-2&-8\ \text{m}\\5\ \text{m}&-10\ \text{m}&2&8\ \text{m}\\\end{array}

Read orientation from the sign of height

A negative signed image height means the drawn image is inverted relative to the object. A positive signed image height means the drawn image is upright.

h_{\text{image}}<0\Rightarrow \text{inverted},\qquad h_{\text{image}}>0\Rightarrow \text{upright}