• the key fact, in one picture: a point in one view does **not** pin its match to a point in the other — it constrains it to a **line**. Given a pixel in the left image, its 3-D point lies somewhere along a single ray; that ray projects, in the right image, to a line — the **epipolar line**. So matching is a **1-D search along a line**, not a 2-D search (this is exactly why we *rectify* stereo pairs — to make those lines horizontal). [fig-epipolar-line]
• **derive the matrix straight from that picture** — ray → plane → line → algebra:
• **(1) the ray (depth ambiguity)**: a pixel back-projects to a ray; in **calibrated** coords ($\hat{x}=K^{-1}x$) the 3-D point is $X=\lambda\hat{x}$, depth $\lambda$ unknown.
• **(2) the epipolar plane**: that ray together with the **second camera center** spans a plane (equivalently: the plane through both camera centers and $X$ — independent of $\lambda$).
• **(3) the epipolar line**: the plane meets the second image in a line, so the match $\hat{x}'$ must lie on it.
• **(4) write the plane in 3-D**: with relative pose $(R,t)$ ($X\mapsto RX+t$; the first center maps to $t$ = the **baseline** vector, the ray direction maps to $R\hat{x}$), the three vectors $\hat{x}'$, $t$, $R\hat{x}$ are **coplanar** → vanishing triple product $\hat{x}'\cdot(t\times R\hat{x})=0$ → $\hat{x}'^{\mathsf T}[t]_\times R\,\hat{x}=0$ → the **essential matrix** $E=[t]_\times R$.
• **(5) → the fundamental matrix**: in raw pixel coords, $x'^{\mathsf T}Fx=0$ with $F=K'^{-\mathsf T}[t]_\times R\,K^{-1}=K'^{-\mathsf T}EK^{-1}$; $Fx$ is the epipolar line. The relative pose $(R,t)$ (translation only up to scale) is recoverable from $E$ — the seed of multi-view reconstruction.
• **forward-ref** (deferred to Advanced/3-D part): *estimating* F/E from noisy matches (the **8-point algorithm**, RANSAC), rectification, triangulation, and **structure-from-motion / multi-view stereo**.
