Derive an image plane by composing ray transformations

1. Establish the optical mechanism

A paraxial ray state is the column (y,θ), where y is signed height and θ is a small signed angle. Free propagation through length l maps y to y+lθ and keeps θ, giving P(l)=[[1,l],[0,1]]. A thin lens of focal length f changes angle to θ−y/f, giving L=[[1,0],[−1/f,1]]. Units matter: the off-diagonal entries are length and inverse length.

For object distance a and sensor distance b, apply the matrices in travel order: M=P(b)L P(a). Multiplication gives A=1−b/f, B=a+b−ab/f, C=−1/f and D=1−a/f. The determinant is one in this same-index ray convention, a useful independent consistency check.

Final height is Ay+Bθ. Rays from one object point have different starting angles, so they meet at one image height only when B=0. Solving gives b=fa/(a−f) and magnification A=−b/a. A sensor placed elsewhere retains angle-dependent spread; selecting one ray does not demonstrate focusing.

The original packet uses f=10 cm, object height 0.1 cm and three angles −0.02,0,+0.02 rad. It reports a local tanθ/θ discrepancy at the largest predicted output angle to illustrate one small-angle approximation. That is not a complete aberration bound: finite thickness, aperture clipping, diffraction and actual lens design remain omitted.

2. Work the original evidence

At a=30 cm and b=20 cm, M=[[-1,−10],[-0.1,−2]] and determinant is 1. The true paraxial image plane is 15 cm with magnification −0.5. At the selected sensor the three rays span 0.4 cm rather than one image point.

3. Predict and investigate

Which matrix entry must vanish for a common object point to image independently of the initial ray angle?

Propagate all three rays explicitly through object-to-lens, lens deflection and lens-to-sensor stages. Compare with matrix multiplication and then solve B=0 without claiming diffraction-free real imaging.

Primary mechanism context. Checked 31 October 2026. Original prose, records and diagrams; no copied source exercises or physical light-source operation. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.