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Epipolar geometry describes the projective relationship between two camera views. A scene point away from the camera baseline and the two camera centers determine an epipolar plane. Its intersections with the image planes are corresponding epipolar lines. Knowing one image point therefore constrains its partner to a line rather than the whole second image.^[ref]

With image points in homogeneous coordinates, a fundamental matrix $F$ expresses the constraintFor an ideal horizontally rectified pair this reduces to equality of image-row coordinates, while horizontal displacement remains free and carries stereo disparity. A line constraint does not determine depth: many scene points along one viewing ray satisfy the geometry. Correct stereo correspondence selects a partner, and camera calibration then supports metric stereoscopic triangulation.

An epipolar line is the image-plane trace of an epipolar plane. Given one point in the first camera image, the partner of that scene point must lie on the corresponding epipolar line in the other view. A fundamental matrix maps the first point to coefficients of that line.^[ref]

For example, let $q_L=(100,50,1{)}^{\mathsf{T}}$ andThen $F{q}_L=(0,-1,50{)}^{\mathsf{T}}$, so a candidate $q_R=(u,v,1{)}^{\mathsf{T}}$ satisfies $-v+50=0$. The partner can have different horizontal coordinate $u$, but must lie on row 50 in this ideal model. Intersecting candidate matches with this line reduces the search; it does not resolve repeated textures or guarantee that a visible partner exists.

An epipolar plane contains both camera centers and a scene point. For a scene point away from the camera baseline, these three points determine a unique plane. Its intersection with each image plane is an epipolar line, so corresponding image points lie on the two traces of the same plane.^[ref]

To see the geometry directly, take camera centers $C_L=(0,0,0)$ and $C_R=(B,0,0)$, and a scene point $P=(X,Y,Z)$. The plane they span satisfies $Zy-Yz=0$. For parallel forward-facing image planes at common focal length $f$, its traces have $y=fY/Z$ in both views. This is why an ideal horizontal stereo pair places corresponding image points on the same row.

Moving the scene point within that plane leaves its epipolar lines unchanged. A single trace therefore cannot recover depth. If the scene point lies on the baseline, the three defining points are collinear and no unique epipolar plane is determined. These degeneracies explain both the usefulness and the limits of line-based stereo correspondence.

The fundamental matrix $F$ encodes the epipolar geometry of two distinct camera centers using image homogeneous coordinates. For matching image points $x_L,x_R$, the ideal constraint isThus $F{x}_L$ is an epipolar line in the right image, while $F^{\mathsf{T}}{x}_R$ is an epipolar line in the left image. A nondegenerate fundamental matrix has matrix rank two and is defined only up to nonzero scale.^[ref]

For an ideal horizontally aligned stereo pair with matched image coordinate origins, one possible scale convention isPutting $x_L=(20,50,1{)}^{\mathsf{T}}$ gives $F{x}_L=(0,-1,50{)}^{\mathsf{T}}$, so the right point must satisfy $v_R=50$. Its horizontal coordinate remains free: the constraint reduces the search to a line but does not determine stereo disparity or depth. Repeated texture can still make stereo correspondence ambiguous.

With camera calibration matrices $K_L,K_R$ and relative pose $(R,t)$ under the convention $X_R=R{X}_L+t$, the relation is $F=K_R^{-\mathsf{T}}[t{]}_{\times }R{K}_L^{-1}$, where $[t{]}_{\times }v=t\times v$. Translation direction affects the geometry, but multiplying $t$ by a scalar only rescales $F$: the fundamental matrix alone cannot establish a metric camera baseline. Pure rotation has $t=0$ and requires a different geometric description.^[ref]

Stereo correspondence identifies image positions in two views that depict the same scene point. This is the matching step needed before stereoscopic triangulation can recover that point. Epipolar geometry restricts a candidate match to an epipolar line; after horizontal stereo rectification, that search follows a corresponding image row.^[ref]

For example, matching a corner at horizontal coordinate 120 in the left view with coordinate 96 in the right gives stereo disparity 24 pixels. A repeated window pattern might offer equally plausible candidates at 96 and 72, producing different inferred depths. Geometry alone does not choose the correct window. A foreground object can also hide the feature in one view: occlusion in stereo vision then prevents a visible match. Texture, agreement of nearby matches and explicit rejection of ambiguous locations supply evidence beyond mere proximity along a row. A confident disparity map must represent locations where stereo correspondence failed as well as those where it succeeded.

Stereo rectification transforms a pair of images so that corresponding epipolar lines become aligned scan lines. In a horizontal stereo arrangement, a scene point visible in both rectified views should have the same vertical image coordinate. This turns a two-dimensional matching search into a one-dimensional search along a row.^[ref]

For example, a feature initially at left row 100 and right row 104 can be mapped to the same row in virtual aligned views. Camera calibration supplies geometry for the transforms, and image resampling evaluates the transformed images on a new pixel grid. The new images need not retain every border region. Stereo rectification aligns the geometric search; it does not identify the correct partner among repeated patterns or restore a feature hidden in one view. The rectified focal length and coordinate offsets must also be used when converting stereo disparity to depth.