# Vanishing points from three collinear points

I would like to find the 2D vanishing point from a three collinear points as is shown in "Multiple View Geometry in Computer Vision" Example 2.19 (see here).

What I did so far:

1 - I've extracted three collinear points (a', b', c') from an image (chessboard) and calculated the distance between these points d(a',b') and d(a',c')

2 - Rewriting the line as points 0, a' and a+b and the known distances in the chessboard 0, a and a+b. I've calculated the 1D homography matrix ($H_{2x2}$) mapping the world points to the image points.

3 - Then, I've calculated the image of a point at infinity with $x' = H_{2x2}(1,0)^T$

So, what I have now is the point at infinity for P$^1$. The question is how to convert this point to the 2D (P$^2$) in order to recovery the affine properties of my image.

Can anyone help me?

$$x′=H_{2x2}(1,0)^T$$ (which you already calculated) is the distance in $$2D$$ of the vanishing point from $$a'$$. The point is along the direction of $$a'b'$$ at a distance $$x'$$ from $$a'$$.