*Projective space*
is invariant under the *n*(*n*+2)parameter *projective group* of
matrices up to
scale (8 dof in 2D, 15 dof in 3D). The fundamental projective
invariant is the cross ratio which requires four objects in a
projective pencil (one parameter configuration, or homogeneous linear
combination of two basis objects
). Projective
space has notions of subspace incidence and an elegant duality between
points and hyperplanes, but no notion of rigidity, distance, points
`at infinity', or sidedness/betweenness. It is the natural arena for
perspective images and uncalibrated visual reconstruction.

*N*-dimensional *affine space* is invariant under the *n*(*n*+1)parameter *affine group* of translations and linear deformations
(rotations, non-isotropic scalings and skewings). Affine space is
obtained from projective space by fixing an arbitrary hyperplane to
serve as the hyperplane of points `at infinity': requiring that this
be fixed puts *n* constraints on the allowable projective
transformations, reducing them to the affine subgroup. The fundamental
affine invariant is ratio of lengths along a line. Given 3 aligned
points *A*,*B*,*C*, the length ratio *AB*/*AC* is given by the cross ratio
*A*,*B*;*D*,*C*, where *D* is the line's affine point at infinity. Affine
space has notions of `at infinity', sidedness/betweenness, and
parallelism (lines meeting at infinity), but no notion of rigidity,
angle or absolute length.

*Similarity* or *scaled Euclidean space* is invariant under
the
*n*(*n*+1)/2+1 parameter *similarity group* of rigid motions
(rotations and translations) and uniform scalings. The fundamental
similarity invariants are angles and arbitrary length ratios
(including non-aligned configurations). Euclidean space is obtained
from affine space by designating a conic in the hyperplane at infinity
to serve as the *absolute conic*. The similarity group consists of
affine transformations that leave the
*n*(*n*+1)/2-1 parameters of this
conic fixed. Angles can be expressed using the cross ratio and
properties of this conic. Scaled Euclidean space has all the familiar
properties of conventional 3D space, except that there is no notion of
scale or absolute length.

Fixing this final scale leaves us with the *n*(*n*+1)/2 parameter
*Euclidean group* of rigid motions of standard *Euclidean space*.