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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.