Points on a line in 3 space through a point
and with
direction
can be written as
. As
varies from
0 to
,
varies from
to the
the point at infinity
.
Under a projective camera
the points project as
where
is the image of
.
The vanishing point of the line,
, is obtained as
or
Thus the vanishing point back projects to a ray with direction
.
Parallel planes in 3 space intersects
in a common
line, and the image of this line is the vanishing line of the plane.
Thus, we have
The set of planes perpendicular to the direction
in the
camera's Euclidean frame have vanishing line
.