The geometry of two eyes. A point in one image does not float free in the other — it is pinned to a single line. The fundamental matrix F is that pinning: for corresponding points x and x′, the epipolar constraint x′ᵀ F x = 0 holds, and F x is the line in the second image the match must lie on. Stereo search collapses from a 2D hunt to a walk along one line. Rendered, not quoted.
source Longuet-Higgins, H.C. (1981), A computer algorithm for reconstructing a scene from two projections, Nature 293:133–135, doi:10.1038/293133a0 (essential matrix) · Faugeras & Luong (1992), the fundamental matrix for uncalibrated cameras. AMBER: primary text paywalled; the geometry below is reconstructed from a constructed camera pair, not quoted.
Two calibrated cameras (K = I). Camera 1 at the origin, P₁ = [I | 0]. Camera 2 rotated and translated, P₂ = [R | t]. A world point X images as the ray x = X in view 1 and x′ = R·X + t in view 2.
The essential / fundamental matrix is the skew of the baseline composed with the rotation:
Because [t]× is rank 2, F is rank 2 by construction — det F = 0. No fitting, no noise; an exact algebraic object.
This is the hinge of two-view geometry. The epipolar constraint x′ᵀFx=0 reduces stereo correspondence from searching a whole image to a 1D search along one line — the basis of structure-from-motion and every multi-view reconstruction pipeline.
Neighbour in THE FOLD: the calibrated case is Longuet-Higgins’ essential matrix E = [t]×R; decompose E and you recover the camera motion (R, t) up to scale — the seed of visual odometry.
Live re-check. Recomputes det F and F·e from the currently active F. Green confirms rank 2 with valid epipoles; flips red the instant window 6 swaps in a full-rank F.
Five 3D points, imaged in both cameras to give correspondences (x, x′):
For each match, F x is the epipolar line l′ in view 2 (blue). The true match x′ (violet) lands on it, and every line passes through the epipole e′ (amber) — the pencil of epipolar lines.
Boot proof, set only after selfcheck passes:
WALL The clean geometry assumes exact, noise-free correspondences and known calibration. Reality breaks each:
• Real matched points carry pixel noise, so the estimated F is almost never exactly rank 2 — you must project it back (SVD, zero the smallest singular value) or lines miss the epipole.
• A degenerate scene — all points coplanar, or pure rotation with zero baseline — makes F unidentifiable; the 8-point solution is ambiguous.
• Wrong matches (outliers) wreck a least-squares fit; robust estimation (RANSAC) is mandatory in practice.
Disclosed planted void: skip enforcing det F = 0 by swapping in a full-rank F (add 1 to the (3,3) entry). The epipolar lines no longer share one epipole and the constraint is only approximately met. The Witness in window 7 catches it live.