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.
The general case of a collineation
is:

with
.
Provided
and
,
this can
be rewritten in inhomogeneous affine coordinates as:

**Property**: *A translation in affine space corresponds to a collineation leaving each
point at infinity invariant.*
*Proof:* The translation
can be represented by the matrix:

Obviously
.
More generally, any affine transformation is a collineation, because it can be
decomposed into a linear mapping and a translation:

In homogeneous coordinates, this becomes:

**Exercise 2.1**
:
Prove that a collineation is an affine transformation if and only if it
maps the hyperplane at infinity *x*_{n+1}=0 into itself (*i.e.* all
points at infinity are mapped onto points at infinity).

*Bill Triggs*

*1998-11-13*