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Stereoscopy supplies depth because different cameras see different projections of the same track. A functional reconstruction first matches the same interaction vertex and track segments across views. A calibrated lens and window model maps each measured image coordinate back to a ray inside the liquid. Refraction at the windows must be included. Intersecting the rays, or finding their closest agreement when measurements have errors, estimates three-dimensional coordinates. Several points then determine a trajectory, rather than only a projected arc.

An original ideal-camera model makes the geometry explicit. Take two parallel pinhole cameras with centers $(0,0,0)$ and $(b,0,0)$, focal length $f$, and optical axes along $z$. Use virtual image planes in front of the pinholes so that image coordinates have the same orientation as the scene. A point $(X,Y,Z)$ then givesSubtracting the horizontal image coordinates gives disparity $d=u_L-u_R=fb/Z$, henceFor stipulated $b=0.20\mathrm{m}$, $f=0.050\mathrm{m}$, $u_L=0.010\mathrm{m}$, $u_R=0.005\mathrm{m}$ and $v_L=0.002\mathrm{m}$, the reconstructed point is $(0.40,0.08,2.0)\mathrm{m}$. These are ideal optical inputs, not dimensions of Alvarez's apparatus. This model omits refraction and lens distortion; actual calibration replaces its simple ray geometry. Since $|dZ/dd|=fb/d^2$, the same disparity error gives worse depth precision for distant points.

The Lorentz force then supplies a momentum scale. With magnetic field $B$ uniform and negligible stopping power over a fitted segment, bubble-chamber track readout gives $p_{\perp }=|q|Br$. The local three-dimensional helix tangent supplies the missing direction: if $\alpha$ is its angle to the field, $p=p_{\perp }/\sin \alpha$. In practical units,Here $e$ is the elementary charge and $c$ the speed of light. A short, almost straight arc poorly constrains its radius of curvature; a visually sharp track can therefore have poor momentum precision.

Stereoscopy enables three-dimensional track reconstruction.^[ref] Calibrated camera positions and optical geometry are needed before a photographed arc becomes a physical radius of curvature. Neutral particles do not directly leave the same continuous ionization trail as charged particles; an interaction or decay can instead reveal them through charged secondary tracks. A photon's inferred path between interactions is not a photographed luminous line.

In a uniform magnetic field, with negligible stopping power and elastic scattering over the fitted segment, the transverse projection is a circle. The Lorentz force turns transverse momentum at angular rate $v_{\perp }/r$, givingHere $r$ is measured in the plane perpendicular to the magnetic field. The relation remains valid with relativistic physical momentum; it does not require $p=mv$. Thus curvature determines magnetic rigidity, not mass or kinetic energy alone. For chosen $|q|=e$, $B=1.5\mathrm{T}$ and $r=0.40\mathrm{m}$, $p_{\perp }\simeq 0.180\mathrm{GeV}/c$. This is an invented example, not a fit to a historical photograph. Reconstructing a three-dimensional helix also determines the component parallel to the magnetic field; the transverse radius of curvature alone does not. Determining the sign of electric charge additionally requires the field direction and the direction of travel, as in the charge-sign ambiguity of a curved track.

The camera center is the center of projection in an ideal perspective camera model: the common origin of its viewing rays. It is a point in scene space, distinct from a principal point, which is an image-plane coordinate. Two distinct camera centers define the camera baseline used by stereoscopy.

For the calibrated perspective projection model $q\sim K(RP+t)$, where $R$ is a rotation matrix, the camera center $C$ satisfies $RC+t=0$. HenceThis follows by applying the inverse rotation to the zero camera-space position.^[ref] For $R=I$ and $t=(-0.1,0,0)$ metres, $C=(0.1,0,0)$ metres: the translation parameter is not directly the world-space camera center. Rotating about that center changes viewing directions without creating a new triangulation baseline.

A depth map assigns a distance-related value to each accepted image location. The geometry must specify whether this means depth along the camera axis or Euclidean distance along the viewing ray. Units, missing values and the coordinate frame are also part of the representation. In calibrated stereoscopy, converting a disparity map through the camera geometry produces depth estimates.^[ref]

For an ideal camera, suppose a point has coordinates $(X,Y,Z)=(1,0,2)$ metres. Its axial depth is two metres, but its Euclidean distance from the camera center is $\sqrt{5}$ metres. The same scene can thus yield different valid numerical maps under different conventions. A missing stereo correspondence requires an invalid-location marker rather than a fabricated depth. Camera calibration and measurement uncertainty determine how confidently accepted values describe the scene.

Stereo disparity is the difference between the image positions of the same scene feature in two views. For an ideal horizontally aligned pair, use signed horizontal coordinates with matched principal points and defineWith the orientation and camera baseline convention of stereoscopic triangulation, a point in front of the cameras has $d=fB/Z$: stereo disparity measures inverse depth, rather than depth itself. Halving $d$ doubles $Z$ when the camera baseline and focal length stay fixed.

A disparity map assigns this difference to matched image locations. Finding the matches is a separate stereo correspondence problem. A repeated pattern can give several plausible matches; a feature hidden from one camera has no visible partner. A small measured stereo disparity is therefore not, by itself, proof of a distant object: the match and calibration must also be credible.^[ref]

The ideal relation also explains why far-away reconstruction becomes sensitive to measurement error. Differentiating $Z=fB/d$ gives $\mathrm{d}Z/\mathrm{d}d=-fB/d^2$; the first-order propagation of uncertainty is ${\sigma }_Z\simeq fB{\sigma }_d/d^2$. Thus equal disparity errors produce larger depth errors at greater depth. The example and curves under stereoscopy quantify this effect.

A stereomicroscope is an optical microscope that sends distinct views of the same specimen to the two eyes through separate viewing channels. The views originate at slightly different directions, producing stereo disparity that supports stereoscopy. Two microscope eyepieces alone do not establish stereo imaging: splitting one image into identical copies supplies no second viewpoint.

A two-objective arrangement uses separate microscope objective assemblies whose viewing directions converge on the specimen. A shared-objective arrangement can instead preserve two distinct viewing bundles through one main objective. The distinction concerns how the two perspectives are formed, rather than the number of eyepieces. A primary manufacturer manual gives a concrete two-path example: its S-series optics have a10° convergence angle and a compact lower assembly providing access around the specimen.^[ref]

For an original geometric illustration, take two viewing rays inclined by $\pm \alpha$ to the normal of a reference plane. A feature at height $h$ above that plane has ray intersections displaced in opposite directions by $h\tan \alpha$, so their separation isWith total convergence angle $2\alpha =10^{\circ }$ and $h=1\mathrm{mm}$, the separation is about0.175mm. This back-projected reference-plane geometry illustrates stereoscopic triangulation; it is not a specification of human depth accuracy or a universal pixel-to-height calibration. Actual image coordinates also depend on each channel's optics and alignment.

The distinct viewpoints are useful when positioning a fine tool near a surface, inspecting a small mechanical assembly or distinguishing raised features from flat markings. Illumination reflected from an opaque surface makes many such specimens accessible without preparing a thin transmitting section. The microscope working distance determines how much physical room remains between the instrument and the focused specimen, which matters when a probe or tool must enter that space.

For measurement, retain the magnification setting and calibrate the relevant channel with a stage micrometer. A scale calibration in a reference plane does not automatically convert stereo displacement into height; that requires the viewing geometry or a separate height calibration. Increasing magnification also does not guarantee improved diffraction-limited resolution. The purpose of the second view is geometric information, alongside the instrument's ability to resolve and magnify structure.

Stereoscopy uses images from different viewpoints to convey or recover depth. In scientific photography, identifying the same feature in two calibrated views permits stereoscopic triangulation: intersecting the directions inferred from the two images.^[ref]

Consider ideal cameras with parallel axes, separated by a baseline $B$. With matching forward-projection coordinates and common focal length $f$ in image-coordinate units, a point at $(X,Z)$ projects toSubtracting gives the stereo disparity $d=x_L-x_R=fB/Z$, hence $Z=fB/d$. If $f$ and $d$ are measured in pixels, $Z$ has the same length unit as $B$.

For an original example, $B=0.12\mathrm{m}$, $f=800$ pixels and $d=24$ pixels give $Z=4\mathrm{m}$. First-order propagation of uncertainty gives ${\sigma }_Z\simeq fB{\sigma }_d/d^2$. Thus fixed disparity precision produces increasing depth uncertainty for more distant objects. Correctly matching the same feature is also essential; ambiguous or hidden features can defeat the reconstruction even when the geometry is known.