Projective Geometry
Hu Zhan Yi
Entities At Infinity
The ordinary space in which we lie is
Euclidean space. The parallel lines usually
do not intersect. If let the parallel lines
extend infinitely, we have vision of their
intersecting at a point, which is a point at
infinity.

There is a unique point at infinity on any a
line.

All points at infinity in a plane make up of a
line, which is the line at infinity of the plane.

All points at infinity in space make up of a
plane, which is the plane at infinity.
Projective Space
With no differentiation between finite points
and infinite points, n-dimensional Euclidean
space and the entities at infinity make up of
a n-dimensional projective space.
Homogeneous Coordinates
In order to study the entities at infinity,
homogeneous coordinate is introduced.
After setting up a Euclidean coordinate
system, every finite point in n-dimensional
space can be represented by its coordinate
(m1 ,..., mn ) . Let x1 ,..., xn , x0 be any
scalars that satisfying:
xn
x1
x0  0,
 m1 ,...,
 mn .
x0
x0
Then ( x1 ,..., xn , x0 ) is called the
homogeneous coordinate of that point.
Relative to homogeneous coordinate,
(m1 ,..., mn ) is called non-homogeneous
coordinate of that point.
Vectors
( x1 ,..., xn ,0)
are defined to be the homogeneous
coordinates of points at infinity.
Projective Parameter
For a line in any dimensional projective
space, any points P on it can be linearly
generated by two fixed points P1 , P2 on it:
X  c1 X1  c2 X 2
where X , X1 , X 2 are the homogeneous
coordinates of P, P1 , P2 respectively, c1 , c2
are two scalars that are not both zero.
c1
c2
The ratio
is called the projective
parameter of P with respect to P1 , P2 on
the line through them.
By allowing c2  0 , the projective
parameter is  .
Cross Ratio
For four collinear points P1 , P2 , P3 , P4 , the
ratio
(1   3 )( 2   4 )
( 2   3 )(1   4 )
is called the cross ratio of ( P3 , P4 ) with
respect to ( P1, P2 ) , denoted by ( P1 , P2 ; P3 , P4 ) .
Where  i are the projective parameters of
Pi , i  1..4 .
Projective Transformation
Let S n , S n' be two n-dimensional
projective spaces , T be a 1-1 map from
S n to S n' . If T preserves:
(i) the incidence relations of points and lines;
i.e. relations: a point on a line, a line
through a point, et.al.
(ii) the cross ratio of any four collinear points,
then T is called a n-dimensional
projective transformation.

The two projective spaces S n , S n' may be
identical.

A n-dimensional projective transformation
can be represented by a (n+1)-(n+1) matrix:
 x1' 
x
   t11 , , t1( n 1)  1 
 
  


 
 '   
 xn 
 xn   t
( n 1)1 ,  , t ( n 1)( n 1) 


'
x 
x
 0
 0
For example: the following map from Pi on
the line L to Pi ' on the line L' is a
1-dimensional projective transformation:
Projective Geometry
Projective Geometry is the geometry to
study the properties in projective space that
is invariant under projective transformation.
Harmonic Relation
We say that the pairs of points ( P1, P2 ) and
( P3 , P4 ) are harmonic if
( P1 , P2 ; P3 , P4 )  1
The pairs of ( P1, P2 ) and ( P3 , P4 ) are
harmonic if and only if
(1   2 )( 3   4 )  2(1 2   3 4 )
where  i are the projective parameters of
Pi , i  1..4 .
Conic
A conic is the totality of points in a
projective plane whose homogeneous
coordinates ( x1 , x2 , x3 ) satisfy the following
equation:
3
a
i , j 1
ij
xi x j  0
(aij  a ji )
where at least one of a ij is nonzero.
The above equation in the definition of a
conic has the equivalent form:
 a11 a12 a13  x1 

 
x1 x2 x3  a21 a22 a23  x2   0
 a a a  x 
.
 31 32 33  3 
The matrix (aij ) is symmetric, and its rank
is not changed under a projective
transformation.
If the determinant of (aij ) is zero, then the
conic is two lines or one line, called
degenerate conic.
Circles, ellipses, hyperbolas and parabolas
are all non-degenerate conics.