Epipolar line from epipole and image point
I have to determine the epipolar line using this model:
I read some books and the Wikipedia-article. But I cannot figure out, what this means:
l2 = e2 x x2
Where l2 ist the epipolar line in the right/2nd image (red line) and x2 is the image point in the right image of the object x.
My problem: As I assume, the point e2 and x2 are in the right image plane, which means, that the cross product of them (in the formula l2) is perpendicular to the image plane and hence cannot be in the image plane, as the red line is.
What am I understanding wrong?
You can write the equation of a straight line in 2D,
ax + by + c = 0
as a dot product
l . x = 0
where l = [a b c]' is the line and x = [x y 1]' is a point on the line.
So, l and x are orthogonal.
In the second image, both e2 and x2 should lie on the epipolar line l2, meaning
l2 . e2 = 0, l2 . x2 = 0
So, l2 is orthogonal to both e2 and x2. You can find a vector that is orthogonal to both e2 and x2 by taking their cross product. Therefore, we can say l2 = e2 x x2.
You can see that l2 . e2, and l2 . x2 are indeed 0, using triple product properties.
l2 . e2 = e2 . l2 = e2 . (e2 x x2) = x2 . (e2 x e2) = 0