Let m1…mn be
3D model points. Let i1…in be 2D image
points Let P be the
plane spanned by m1, m2, and m3. Let m4’,… mn’ be the projection of m4…mn onto P. Write the
affine coordinates of mi’ relative to m1,m2,m3, as (ai,bi).
The image
points depend on the viewing direction.
For some viewing direction, let i’j be
the intersection of the line connecting
mj and ij with P.
The triangle
formed by m4,m4’,i’4 is similar to the one formed by mj,mj’,i’j, for any j. So we have:
(m’4-i4’) =
rj(m’j-ij’) for some scale factor rj.
i’j appears
in the same image position as mj.
Since i’j is coplanar with p1,p2,p3,
it’s affine coordinates are invariant to
projection. They are (aj,bj).
Then,
writing m’j, i’j with affine coordinates we have:
(aj,bj)
= (aj,bj) + ((a4,b4) – (a4,b4))/rj.
Looking at
either component, we get a series of linear equations which define a line in alpha or beta space.