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Compute Epipolar Line

Given a fundamental matrix $F$ and a point $\mathbf{x}_1 = [x, y, 1]^T$ in image 1, compute the corresponding epipolar line $\mathbf{l}_2$ in image 2.

The epipolar line is computed as:

$\mathbf{l}_2 = F \cdot \mathbf{x}_1$

where $\mathbf{l}_2 = [a, b, c]^T$ represents the line $ax + by + c = 0$ in image 2.

Normalization: The line coefficients should be normalized so that:

$\sqrt{a^2 + b^2} = 1$

This ensures the coefficients represent a proper line equation where $c$ gives the signed distance from the origin to the line.

Epipolar lines are fundamental in stereo vision -- they constrain where a corresponding point can appear in the second image, reducing the search from 2D to 1D.

Round each coefficient to 4 decimal places.

Example:

Input:

F = [[0, 0, 0],
     [0, 0, -1],
     [0, 1, 0]]
x1 = [10, 20, 1]

Output:

[0.0, -1.0, 20.0]

Reasoning:

First, we compute the epipolar line $\mathbf{l}_2$ by multiplying the fundamental matrix $F$ with the point $\mathbf{x}_1 = [10, 20, 1]^T$ : $\mathbf{l}_2 = F \cdot \mathbf{x}_1 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 10 \\ 20 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \\ 20 \end{bmatrix}$
Then, we normalize the line coefficients $[0, -1, 20]^T$ to satisfy $\sqrt{a^2 + b^2} = 1$ : since $\sqrt{0^2 + (-1)^2} = 1$ , the coefficients are already normalized.
The final output is $[0.0, -1.0, 20.0]$ after rounding each coefficient to 4 decimal places.

Constraints:

F: 3x3 fundamental matrix as list of lists
x1: Point in image 1 as [x, y, 1] (homogeneous coordinates)
Return: Normalized line coefficients [a, b, c] as a list
Normalize so that sqrt(a^2 + b^2) = 1
Round to 4 decimal places

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.

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