Ideal points on the projective plane are located at infinity, and have coordinates of the
form . There is just one free parameter in the coordinates of an
ideal point because the scale of a homogeneous vector is
arbitrary - thus the set of all ideal points on the projective plane
constitutes a line, called the * ideal line*. In the same way, the
ideal points of projective 3-space have the form and
constitute
a plane called the ideal plane, and so on for projective spaces of any
dimension.

The image of an ideal point under a projectivity is
called a * vanishing point*, the image of an ideal line is called a
* vanishing line*, and so on. A familiar example
of the formation of a vanishing point is shown in Figure
4. A camera with focal point **C** views a scene in which
there are a pair of parallel lines **M** and **N**. Lines **M** and **N**
appear in the
image as converging lines **m** and **n**
which meet at the vanishing point **V**. An
important feature which is discussed below is that the line **CV**
is * parallel* to lines **M** and **N**.

**Figure:**
* The camera centre is at C. The parallel lines M and
N in the scene give rise to converging lines m and n
in the image which
meet at the vanishing point V. The line CV is parallel to
the lines M and N, with direction vector d.
*

An algebraic analysis of this configuration would describe the
following - lines **M** and **N** are parallel and so meet at
infinity at an ideal point; the image of **M** and **N**
meeting at the ideal point is **m** and **n**
meeting at the
vanishing point **V** (the image of the ideal point is **V**) -
note that concurrency is preserved under a
projective transformation, so concurrency at the ideal point in the
world is reflected in concurrency at **V** in the
image; finally, the ray **CV** must
itself be incident with the ideal point because this is perspective
projection, and hence the ray **CV** is parallel to
lines **M** and **N** in the scene.
The analysis would involve some complication, because
lines are more difficult to manipulate in projective 3-space than they
are on the projective plane. In fact, an algebraic analysis is
unnecessary to gain a reasonable understanding of the physical
configuration, which can be obtained instead by a straightforward
geometric argument as below.

**Figure:**
* Two planes constructed in the way described in the text. By the
arguments in the text, C and V are both on the line of
intersection of the two planes, and the line of intersection has
direction d (see Figure 4). Thus, CV is
parallel to the lines M and N in the scene in Figure 4.
*

Referring again to Figure 4, consider a plane
through **C** and line **M**, and a plane through **C** and
line **N**. We now show that the line of intersection of the two
planes has the same direction ** d** as lines **M** and **N** in the
scene, and that **CV** coincides with the line of intersection of the
two planes i.e. the line **CV** is parallel to the lines in the scene.
See Figure 5.

First note that the line of intersection
of and has direction
because both planes and have as a basis
vector. Now note that
intersects the image plane along **m**,
and similarly intersects the image plane along **n**. Thus,
the line of intersection of and
passes through the point of intersection of **m** and **n** which is the vanishing point **V**.
The line of intersection of and also passes
through **C**, because **C** is incident with both planes.
Thus, **CV** coincides with the line of intersection of and
, and the argument is concluded.

The discussion can be extended to the imaging of a plane, as shown in
Figure 6. A camera with focal point **C** views a
tiled plane .
One set of parallel lines on is concurrent at the ideal point
, and the other set is concurrent at . The line through
and is the ideal line (the horizon) of the plane.
The image of is the vanishing point , the image of is
the vanishing point , and the image of the ideal line is the
vanishing line **L**.
An important feature of this configuration, which can
be demonstrated using similar arguments to those above, is that
the normal of the plane defined
by **C** and **L** is * parallel* to the normal of the
plane - see Figure 6b.

**Figure:**
*
(a) One set of parallel lines on the plane is concurrent at the ideal point
, and the other set is concurrent at . The line through
and is the ideal line (the horizon) of the plane.
The image of is the vanishing point , the image of is
the vanishing point , and the image of the ideal line is the
vanishing line .
(b) Note that
the plane formed by and L has the same normal as the plane .
*

Fri Nov 7 12:08:26 GMT 1997