= Bubble-chamber stereo reconstruction
<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 gives
$$
u_L=\frac{fX}{Z},\qquad u_R=\frac{f(X-b)}{Z},\qquad
v_L=v_R=\frac{fY}{Z}.
$$
Subtracting the horizontal image coordinates gives disparity $d=u_L-u_R=fb/Z$, hence
$$
Z=\frac{fb}{d},\qquad X=\frac{Zu_L}{f},\qquad Y=\frac{Zv_L}{f}.
$$
For 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.
\Image[/hydrogen-bubble-chamber-stereo-reconstruction.png]
{title=From stereo bubble photographs to coordinates, momenta and an event ensemble}
{description=Original functional diagram. The chamber, camera positions and tracks are schematic, not a historical mechanical drawing. The parallel-camera ray construction illustrates an idealized model without refraction.}
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,
$$
\frac{p_\perp}{\mathrm{GeV}/c}=0.299792458\left|\frac qe\right|\frac B{\mathrm T}\frac r{\mathrm m}.
$$
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.
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