I'm trying to test my understanding by deriving famous results from computer vision. This one: the essential matrix.
I'm using the scalar triple product to encode the co-planarity constraint.
I'm assuming the camera center of projection that corresponds to $x_0$ is the origin, and the rotation for the first camera is the identity matrix. I'm assuming the rotation and translation of the second camera with respect to the first is $R$ and $T$ respectively.
The strategy is to derive three vectors from this setup, and then use the scalar triple product to get the essential matrix.
Here's my math:
$$A \cdot (B \times C) = 0$$ $$A = R^{-1}x_1 = R^{T}x_1$$ $$B = T$$ $$C = x_0$$ $$A = R^{T}x_1 \cdot [T]_{x}x_{0}$$ $$A = x_1^{T}R[T]_{x}x_{0}$$ $$A = x_1^{T}Ex_{0}$$ $$E = R[T]_{x} $$
The correct answer is $E = [T]_{x}R$. If I had to guess, I think I may have account for the translation in the $A$ vector somehow. But I'm not sure, since it's a direction only -- it should still encode the co-planarity constraint? Please advise.