Lens Equation Inference Scans

Signed Height Linearity

scale the object, scale the image

With lens geometry fixed, signed magnification turns object height into signed image height row by row. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Keep the lens geometry fixed

Use the same lens placement for every row: object distance 15 metres and image distance 30 metres. That fixed geometry makes the signed magnification stay at -2.

m=−vu=−2m=-\frac{v}{u}=-2
Fixed lens geometryThe checked rays set the shared magnification.FFlensobjectimage

Image height is linear in object height

Once the geometry fixes magnification, object height is the only remaining input. Doubling object height doubles signed image height; tripling it triples signed image height.

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

Three object heights show the scale

Scan three object heights while the lens stays fixed. The negative sign records inversion in every row, and the size changes in equal steps.

hobjectumhimage1 m15 m−2−2 m2 m15 m−2−4 m3 m15 m−2−6 m\begin{array}{c|c|c|c}h_{\text{object}}&u&m&h_{\text{image}}\\1\ \text{m}&15\ \text{m}&-2&-2\ \text{m}\\2\ \text{m}&15\ \text{m}&-2&-4\ \text{m}\\3\ \text{m}&15\ \text{m}&-2&-6\ \text{m}\\\end{array}
Signed height linearityThe middle row is the checked ray diagram.FFlensobjectimage

The diagram is the middle table row

The rendered object height is the middle row. Its image height is negative, so the image arrow points below the axis while keeping twice the size.

hobject=2 mhimage=−4 mh_{\text{object}}=2\ \text{m}\qquad h_{\text{image}}=-4\ \text{m}
Signed height linearityThe checked image height is the middle table result.FFlensobjectimage