If there are 9 unknowns, why do we only need 8 points?
motoole2
Although there are 9 unknowns, the fundamental matrix $F$ (and the essential matrix $E$) are only unique up to a scale factor. To see this, I can scale the values of $F$ by an arbitrary scalar $\alpha \neq 0$, and show that if
$(x'^T F x) = 0$
then
$x'^T (\alpha F) x = \alpha (x'^T F x) = 0 $
In other words, this constraint on epipolar geometry is unaffected by the scalar $\alpha$. (Remember that $x'$ and $x$ are still homogeneous coordinates too, and their scale also has no affect on the 2D points that they represent.)
So while there are 9 unknowns, there are really only 8 degrees of freedom. And we need 8 correspondences to solve for the fundamental matrix.
If there are 9 unknowns, why do we only need 8 points?
Although there are 9 unknowns, the fundamental matrix $F$ (and the essential matrix $E$) are only unique up to a scale factor. To see this, I can scale the values of $F$ by an arbitrary scalar $\alpha \neq 0$, and show that if
$(x'^T F x) = 0$
then
$x'^T (\alpha F) x = \alpha (x'^T F x) = 0 $
In other words, this constraint on epipolar geometry is unaffected by the scalar $\alpha$. (Remember that $x'$ and $x$ are still homogeneous coordinates too, and their scale also has no affect on the 2D points that they represent.)
So while there are 9 unknowns, there are really only 8 degrees of freedom. And we need 8 correspondences to solve for the fundamental matrix.