Let
and
be the vanishing points of two
lines in the image. If
is the angle between the two
scene lines, we have
If
is known the above equation gives a quadratic constraint
on the entries of
.
If it is known that the scene lines are orthogonal (
),
then we have a linear constraint
Thus, given five pairs of perpendicular lines, one can solve for the
entries of
.
The vanishing point
of the normal direction to a plane
is obtained from the plane vanishing line as
A common example is a vertical direction and a horizontal plane.
Writing the above as
removes the homogeneous scaling factor and results in three
homogeneous equations linear in the entries of
.
Given a sufficient number of such constraints
can be
computed and
follows.
The following can be verified by direct computation:
If
( no skew) then
.
If, in addition,
then
Suppose it is known that the camera has zero skew and that
the pixels are square (or the aspect ratio is known) the
and
can be computed from an orthogonal triad of directions.