Points on the plane at infinity (
), which
may be written as
are mapped
to the image plane by a general camera
as
Thus
is the planar homography between
and the image plane. Note that the mapping is
independent of the position (translation) of the camera and
depends only on the orientation. (An explanation as to why the
images of stars stay fixed on the retinae as we translate?)
Since the absolute conic (
) is on
, we can compute its image as
Proof.
Note that under a point homography
which maps
to
, a conic
is mapped to
. Hence
on
maps to
Like
,
is an imaginary point
conic with no real points. It cannot really be observed in an
image. It is really an useful mathematical device.
depends only on the internal parameters of the
camera and is independent of the cameras position or orientation.
It follows from above that the angle between two rays is given
by the simple equation
The above expression is independent of the choice of the projective
coordinate system on the image. To see this consider any 2D
projective transformation
. The points
are
transformed to
, and
transforms
(as any image conic) to
.
Hence the expression for
is unchanged.
We may define the dual image of the absolute conic as
Once
(equivalently
) is identified
in an image
is uniquely determined; since a symmetric
matrix can be uniquely decomposed into an upper triangular matrix
and its transpose (
) by
Cholesky decomposition.
An arbitrary plane
intersects
in a
line, and this line intersects
in two points (imaginary)
which are circular points of
. The image of the
circular points line on
at the points at which the
vanishing lines of the plane
intersects
.