A *collineation* of
is defined as a mapping from the plane to
itself such that the collinearity of any set of points is preserved. Such a
mapping can be achieved with matrix multiplication by a
matrix
*T*. Each point
is transformed into a point
:

We will use the terms

To transform a line
into a line
,
we note that collinearity must
be preserved, that is, if a point
lies on the line
,
then the
corresponding point
must lie on the corresponding line
.
Therefore,

which indicates that

From these results, it is not hard to show that a point conic *C* transforms
to
*T*^{-T}*CT*^{-1},
and a line conic |*C*|*C*^{-1} transforms to
*T*|*C*|*C*^{-1}*T*^{T}.

Regarding transformations, recall that *projective*
*affine*
*similarity*
*Euclidean*.
Let's study the matrix *T* to uncover the relationships between these
various geometries. First we will write out the elements of *T*, for
reference:

The affine plane is just the projective plane minus the ideal line.
Therefore, affine transformations must preserve the ideal line and
the ideal points,
that is, any point
[*X*, *Y*, 0]^{T} must be transformed into
for some arbitrary scaling
:

which implies that

where once again only six of these parameters are independent, since scale is unimportant.

Unlike affine transformations,
similarity transformations preserve angles and ratios of lengths.
Delaying the derivation for a moment, we simply state the result:

where is an arbitrary angle.

Under Euclidean transformation, scale is important, and therefore the
point
must first be converted to Euclidean coordinates by
dividing by its third element. The transformation then is

In closing this section, we offer one final proposition, along with its proof:

The ``only if'' is rather easy to see: The absolute point is transformed through equation (3) to the point , which is equivalent because the scale factor is ignored.

The ``if'' is a little more complicated, but still rather straightforward.
Starting with the unrestricted equation for T,

the fact that [1,

Since the elements of

So the matrix of

Given two arbitrary numbers

which is seen to be the equation of a similarity transformation when compared with equation (3). (NOTE: Using the point [1,-