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So in the following lecture notes about epipolar geometry, the essential matrix and the fundamental matrix:

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$T$ is the translation vector between the cameras. $P_l$ is the vector from $C_l$ to $P$. My question is, why is:

$$ P_r = R(P_l - T) $$

Here, $R$ is a rotation matrix. However, based on these vectors, I don't see why you need to rotate the vector. In fact, I think that the relationship should just be:

$$ P_r = P_l - T $$

Does anyone know why there is a rotation matrix?

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$T$ is only a translation of the vector. So if you apply $P_l - T$ you get vector $P_l$ at position $C_r$. Since you also want to point to the same direction as $P_r$ you have to use a rotation matrix. Thus, $P_r=R(P_l-T)$.

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