The aim of this text is to introduce the basic concepts of projective geometry in their relations to some important computer vision problems. The idea of using the projective spaces instead of more familiar Euclidean spaces may seem strange at first. However, once the fundamental notions are fully understood, it will be appreciated as a particularly convenient setting for the treatment of some of the important fundamental vision problems, such as the calibration, stereo motion estimation problems. Here, we will first touch upon the basic algebraic geometric concepts. Then the above-mentioned computer vision problems will be overviewed. Our discussion of the fundamental computer vision problems will be in the same line with Faugeras’ book.
The whole idea behind projective geometry is completeness. In R2, parallel lines do not intersect. However intuitively we tend to think that they intersect at infinity and this is what we will have in projective plane P2.
Making the following substitutions
We will call this new representation of the plane, where points are defined
by three coordinates 
which are obtained through the above substitution, the projective plane P2. Furthermore, as seen from its definition lines are homogeneous entities
in the projective plane.
Going back to our line equation (*), our family of parallel lines has a solution with
Two example solutions of this system are 
. In general, it can easily be seen that if 
is a solution then so is 
. Going back to our intersection at infinity intuition, we will call this
system of solutions the point at infinity of the line.
Thus, given another family of parallel lines, say 
we have a different solution we will call the point at infinity of the family.
Apparently, all these infinity points lie on the same line, i.e., x3 = 0 which is called the line at infinity.
Let’s reinforce our insight into the projective notions through another intuitive picture picked up from the principles of perspective drawing which we owe to Renaissance artists. Assume that we want to draw two highways starting close to one another at a station and going in different directions.
The analogy is that a, b are points at infinity of two families of parallel lines (our highways) and the line at infinity is the horizon.
Definition: Projective plane P2 is composed of points x with coordinate vector 
not all zero and defined up to a scale factor.
i.e., 
represent the same point (equivalence classes of points)
1. There is a 1-1 correspondence between the points of the usual affine plane R2 and that of the projective plane P2- {x3=0 line at infinity }. In fact,
2. Given the basis 
(which is called the standard basis). A point can be written as a linear
combination of the first three basis vectors.
3. The projective line P1 and the projective space P3 are defined in a similar manner. The projective line P1 is the point set (x1, x2) not both zero and defined up to a scale and where x2 = 0 is called the point at infinity. The projective space P3 is the set of points 
defined similarly and x4 = 0 is called the plane at infinity.
Let (x, y, z) be the homogeneous coordinates of a point p in P2 then all the other homogeneous coordinates of the same point are given
by 
.Then they lie on a line passing through the origin which is not an element
of the projective plane.
y
z
l(x1, x2 ,x3)
(x1, x2 ,x3)
x
Another interpretation of the same picture is given below. This brings us back to the 1-1 correspondence between the usual affine space R2 and P2 excluding the line at infinity.
Does this picture remind you something? Isn’t it our simple-minded pinhole camera model? Let’s redraw the figure so that some of the important components of our mind’s eye’s camera model become visible.
This overall picture seems to be in agreement with the idea of employing projective thinking for computer vision problems. Here the world coordinate system and the image plane coordinate system overlap (their xy planes are the same). This does not make the model any less general. We will have means to extend this model to obtain different coordinates frames for the two.
There is another interesting interpretation of the projective plane, namely the central projection.
Since (x1, x2 ,x3) is equivalent to (lx1, l x2, lx3). Why not represent each equivalence class by
x = (x1, x2 ,x3) with || x || = 1
This gives rise to the following picture
In more theoretical terms, the projective plane P2 is topologically equivalent to the unit sphere S2 with antipodal points, namely (x, y, z) and (-x, -y, -z) , are identified.
For the projective line the corresponding topologically equivalent object is the unit circle
± ¥ ± ¥
We need to know a couple of other tools which will be operational in our vision problems.
In projective plane a line is charcterized by a triplet (u1, u2, u3). The line equation, as seen before, is given by
From this equation it is easily seen that lines are formally no different from points. A point can be represented in terms of the lines through it (called pencil of lines). Conversely, a line is thought of as the set of points on it (range of points). They are both expressed via the same line equation (*).
In P3, the projective space, the duality is between points (thought of as the
pencils of planes) and planes (thought of as the range of points) where
a plane is defined by a four-tuple 
.The eqution of the plane is given as
For Pn a collineation (linear transformation) is an nxn invertible matrix which transforms a given point in Pn into another point of the same space.
In R2 an affine transformation is defined as
(*)
where A is a 2x2 matrix
Take the projective transformation
Consider transformations (*) which preserve a line (without loss of generality x3=0)
So the conclusion is that affine transformations are those projective transformations preserving the line at infinity.
In 3-D space the same idea gives rise to the following similar correspondence
An Euclidean transformation is defined through an orthogonal rotation matrix A (AAT=I) and a translation vector t. In 2-D, A is conventionally represented as
Clearly (1, ± i)T are the eigenvectors of A. Conversely, if we fix the points (1, ± i, 0), sometimes called the absolute points, on the projective plane this corresponds to
Thus Euclidean transformation are those projective trasformations which preserve (1, ± i, 0). Clearly, the line x3=0 lying between these two point is preserved, too.
In terms of R3 and P3, Euclidean transformation are solid motions which leave a special conic invariant. This conic is called the absolute conic and it is the intersection of the conic
In other words, the intersection of
The absolute conic has a special importance in computer vision. As we will see it will be used as the calibration pattern for determining intrinsic camera parameters as it is preserved under an Euclidean transformation.
The cross-ratio is an invariant quantity that remains constant under a collineation. Besides, Laguerre formula allows us to determine the angles between lines and plane through the use of the cross-ratio. It is a powerful tool very well appreciated by computer vision researchers.
The cross-ratio of four points, denoted by {a,b ; c,d}, is defined as follows
Or alternatively, it is defined as
where e1 and e2 are standard basis vectors for the projective line (e1 = [1,0], e2= [0,1]).
{ l1, l2 ; l3, l4 } = { p1, p2 ; p3, p4 }
l l1
l2
p1
p2 l3
p3
p4 l4
a
Here the line l intersecting all four lines is selected arbitrarily.
l ¥ ja
l2
j
p2 ia
i
a p4 l4
a
If ia and ja are the lines joining the absolute points with the point a, by Laguerre Formula
a= 1/2i log ({ l1, l2 ; ia, ja})
In particular, eip = Cosp + i Sinp = -1. Thus, if the cross-ratio { l1, l2 ; ia, ja} = -1 then l1 and l2 are perpendicular.