Given a coordinate system,
-dimensional real affine space
is the set of all points parameterized by
An affine transformation is expressed as
(usually) non-singular matrix and
vector representing a translation.
In the special case of when
is a rotation (i.e.,
, the the transformation
Transformation of one point (or one axis) completely determines
an Euclidean transformation, an affine transformation in
is completely determined by a mapping of
points (3 points for
It is easy to verify that an affine transformation preserves
parallelism and ratios of lengths along parallel directions. In fact,
coordinates in an affine geometry are defined in terms of these
fundamental invariants. An Euclidean transformation, in addition to
the above, also preserves lengths and angles.
Since an affine (or Euclidean) transformation preserves parallelism
it cannot be used to describe a pinhole projection. We need to
projective geometry to represent such transformations.