Affine and Euclidean geometries

• Given a coordinate system, $n$ -dimensional real affine space is the set of all points parameterized by ${\bf x} = (x_1,\ldots,x_n)^t \in {\mathbb{R}}^n$ .
• An affine transformation is expressed as
  $${\bf x'} = {\bf A}{\bf x} + {\bf b}$$
  where ${\bf A}$ is a $n \times n$ (usually) non-singular matrix and ${\bf b}$ is a $n \times 1$ vector representing a translation.
• By SVD
  $${\bf A} = {\bf U}{\bf\Sigma}{\bf V}^T = ({\bf U}{\bf V}^T)({\bf V}{\bf\Sigma}{\bf V}^T) = R(\theta)R(-\phi){\bf\Sigma} R(\phi)$$
  where
  $${\bf\Sigma} = \left[ \begin{array}{cc} \lambda_1 & 0 \\ 0 & \lambda_2 \end{array}\right]$$

• In the special case of when ${\bf A}$ is a rotation (i.e., ${\bf A}{\bf A}^t = {\bf A}^t{\bf A} = {\bf I}$ , the the transformation is Euclidean.
• Transformation of one point (or one axis) completely determines an Euclidean transformation, an affine transformation in $n$ dimensions is completely determined by a mapping of $n+1$ points (3 points for a plane).
• 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.