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Autocollimation is an optical return-path method: a source pattern is projected through a collimator, reflected back from an external surface, and examined through the same optical system. It is the process used by an autocollimator, rather than the instrument itself. A plane mirror normal to the outgoing collimated light returns the light along the reference direction.^[ref]

A typical apparatus uses an illuminated reticle near a focal plane, an objective and a beam splitter that separates the illumination and viewing paths. The outward projection and the return image are compared, making small changes in reflector orientation observable.^[ref]

For a small mirror tilt $\theta$, reflection changes the return direction by $2\theta$. In the ideal paraxial model, return-image displacement is $x\approx 2f\theta$, where $f$ is the effective focal length. An original example with $f=150\mathrm{mm}$ and $\theta =20\mu \mathrm{rad}$ gives $x\approx 6\mu \mathrm{m}$. This is displacement in the image plane, not a direct measurement of mirror translation.

The angular interpretation explains why simply moving an ideal plane mirror farther away does not multiply the focal-plane displacement by that distance. In practice, finite aperture, beam walk, alignment and diffraction limit the usable return. Measurement calibration and a stable reference remain necessary to distinguish a genuine angular change from movement of the optical system.

An autocollimator is an optical instrument that measures small angular changes by sending collimated light toward a reflector and examining the returning image. The outgoing and returning paths share optical components: a beam splitter directs light from an illuminated reticle through a collimator, while the reflected light is imaged onto a detector or viewed through an eyepiece. The measurement uses autocollimation, the return of the instrument's own projected light.^[ref][ref]

A small tilt $\theta$ of a plane mirror changes the reflected beam direction by $2\theta$. In an ideal paraxial model with focal length $f$, the return-image displacement from its zero position iswith the angle in radians. For the independently chosen values $f=200\mathrm{mm}$ and $x=4\mu \mathrm{m}$, the mirror tilt is $10\mu \mathrm{rad}$, about 2.063 arcseconds. The reflected beam turns through twice that angle: confusing mirror tilt with beam deflection introduces a factor-of-two error.

A mechanical application turns angular readings into height differences. A mirror carried on two supports separated by a base $b$ has height difference $h=b\tan \theta$ across that base. With an illustrative $b=100\mathrm{mm}$ and the tilt above, $h$ is about 1 micrometre. A series of such local slopes can constrain straightness, provided the support geometry and reference are accounted for.^[ref]

The instrument therefore senses orientation through image position. It is not directly a distance gauge: lateral translation of an ideal plane mirror without tilt does not encode that translation in the same angular reading. Zero setting, measurement calibration and optical alignment remain part of interpreting the observation.

A plane mirror sends each incident ray into its specularly reflected direction, with equal incidence and reflection angles measured from the surface normal. For a point object in front of the mirror, the backward extensions of the reflected rays meet at the reflected point behind the plane. The image is therefore virtual, upright, equal in size and as far behind the mirror as the object is in front. The actual rays do not travel behind the mirror. In coordinates with mirror plane $x=0$, the object point $(x,y,z)$ is mapped to $(-x,y,z)$.

A converging convex lens refracts a paraxial pencil according to the thin-lens equationwhere $u>0$ is the distance to a real object, $f>0$ is the focal length, and $v>0$ denotes an image on the opposite side. Two constructed rays locate the image: a ray parallel to the optical axis emerges towards the far optical focus, and a ray through the ideal thin lens's optical centre proceeds undeviated. Their intersection is a real image; if they diverge, their backward extensions locate a virtual one.

For $u>2f$ the real inverted image is reduced; at $u=2f$ it is equal in size; for $f<u<2f$ it is enlarged. At $u=f$ the emergent pencil is parallel and the image is at infinity in the ideal model. For $0<u<f$, the image is virtual, upright and enlarged on the object's side. A distant object gives an image near the far focal plane. The lensmaker's equation explains why shape alone does not guarantee convergence: the surrounding medium and refractive index also matter. Spherical aberration, chromatic aberration and diffraction limit the ideal point-image construction. The statement that the centre ray is undeviated is the thin-lens approximation, rather than a general rule for a thick compound optical instrument.

The image is upright vertically when viewed down from behind, but left and right are interchanged. To track the image formation, let $+x$ point forwards and $+z$ upwards. Incoming light travels along $-x$. The $45^{\circ }$ flip plane mirror sends it along $+z$. Before this reflection, the optical lens has inverted the scene's top into the $-z$ direction; the law of reflection maps that direction into $+x$ on the horizontal screen. The screen's far edge is the top of the picture for someone looking down from behind. The horizontal inversion from the optical lens remains, since this plane mirror does not change the lateral coordinate.

The minimum is two additional plane mirrors, making three including the flip plane mirror. Their reflecting planes can meet at $45^{\circ }$ in the vertical section, as in a pentaprism. For example, in the $xz$ section take the first added reflecting plane at $67.5^{\circ }$ to $+x$ and the second at $112.5^{\circ }$. The upward beam first travels diagonally forwards and upwards, then horizontally backwards towards an eye looking forwards. The two reflections together rotate directions through $90^{\circ }$: the propagation direction $+z$ becomes $-x$, while the screen's top direction $+x$ becomes $+z$ at the eye. Thus the picture remains upright vertically.

One additional plane mirror would also send $+z$ into $-x$, but it would send the top direction $+x$ into $-z$, making the image upside down. This proves minimality for ordinary plane mirrors. The three-reflection arrangement still reverses left and right; correcting that additionally requires a roof arrangement or another suitable image-erecting system. Vertical erection alone is what is asked here.

A plane mirror has a flat reflecting surface. The law of reflection sends incident unit direction $\mathbf{d}$ to $\mathbf{d}-2(\mathbf{d}\cdot \mathbf{n})\mathbf{n}$, where $\mathbf{n}$ is its unit normal. It forms a virtual image of an object the same perpendicular distance behind the reflecting plane as the object is in front. Two successive plane mirrors whose planes meet at angle $\gamma$ rotate the ray geometry through $2\gamma$ about their intersection.