= Image formation
<Image formation> assigns each object point a corresponding image point: outgoing <light rays> from that object point either converge there, giving a <real image>, or diverge as though they originated there, giving a <virtual image>. The requirement applies separately to each point. Bringing unrelated <light rays> together makes a bright patch, but does not preserve the object's pattern. This point-by-point correspondence explains how an <optical camera> records a scene. https://openstax.org/books/university-physics-volume-3/pages/2-4-thin-lenses[Ling, Sanny and Moebs, University Physics Volume 3, §2.4: Image formation by thin lenses]{ref}.
Use the same convention as <thin lens>: place the <optical lens> at $x=0$, with incident <light> travelling to the right and a real object at $x=-u$, $u>0$. Write the image coordinate as $x=v$, so $v>0$ for a <real image> and $v<0$ for a <virtual image>. Heights $h_o,h_i$ are positive above the <optical axis> and negative below it. In air, within the <paraxial approximation>, a <thin lens> obeys
$$
\frac1u+\frac1v=\frac1f,\qquad v=\frac{fu}{u-f},\qquad m=\frac{h_i}{h_o}=-\frac vu.
$$
Here $f$ is the signed <focal length>, positive for a converging <optical lens> and negative for a diverging <optical lens>; the expression for $v$ assumes $u\ne f$. Object distance $u$ is positive even though the object's coordinate is negative. These coordinate rules describe transmission through the <optical lens>; a mirror reverses the outgoing direction, so do not identify its image distance with this same rightward coordinate. https://openstax.org/books/college-physics-2e/pages/25-6-image-formation-by-lenses[Urone and Hinrichs, College Physics 2e, §25.6: Lens equation and sign conventions]{ref}.
The beauty of <geometrical optics> is that two convenient <light rays> can locate a point, while a whole admitted bundle forms its image. For a converging <thin lens>, an incident <light ray> parallel to the <optical axis> exits through the rear <optical focus>; a <light ray> through the centre continues undeviated. Real <optical lens> elements do not focus every admitted <light ray> exactly: <optical aberrations> and <diffraction> limit the point correspondence. The diagrams below are original ideal-model constructions, not measured data. https://openstax.org/books/university-physics-volume-3/pages/2-4-thin-lenses[Ling, Sanny and Moebs, §2.4: Ray tracing and imaging limits]{ref}.
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