Four aligned points provide a projective invariant, the cross ratio.
If we only allow affine transformations, three aligned points are
enough to provide an invariant: the ratio of their separations.
Finally, limiting ourselves to rigid planar motions, there is an
invariant for only two points: their separation. In other words,
invariants are only defined relative to a *group*
of
permissable transformations. The smaller the group, the more
invariants there are. Here we will mainly concentrate on the group of
planar projective transformations.

Invariants measure the properties of configurations that remain
constant under arbitrary group transformations. A configuration may
contain many invariants. For instance 4 points lead to 6 different
cross ratios. However only one of these is *functionally
independent*: once we know one we can trivially calculate the other
five. In general, it is useful to restrict attention to functionally
independant invariants.

An important theorem tells us how many independant invariants exist for
a given algebraic configuration. Let
be the configuration and
dof
its number of degrees of freedom, *i.e.* the number
of independant parameters needed to describe it, or the dimension of the
corresponding parameter manifold. Similarly, let dof
be the
number of independent parameters required to characterize a
transformation of group ;
let
be the subgroup
of
that leaves the configuration invariant as a shape; and
let dof
be this *isotropy subgroup*'s number of
degrees of freedom. Then we have the following theorem
[6]:

Consider a few examples:

- 1.
- Let
be the 2 dof group of affine transformations on the
line. No continuous family of affine transformations leaves 3 collinear
points (which have 3 dof) invariant:
.
So there
is 3-2+0=1 independant invariant,
*e.g.*one of the length ratios. - 2.
- Let
be the 3 dof group of rigid transformations in the plane
and
be the 3 dof configuration of two parallel lines.
is the 1 dof subgroup of translations parallel to this
line, so there is 3-3+1=1 independant invariant,
*e.g.*the distance between the lines. - 3.
- Two conics in the projective plane have dof. The planar projective group has 8 dof and no continuous family of projective transformations leaves two conics globally invariant; therefore there are 10-8+0=2 invariants for such a configuration.

In practice, the isotropy subgroup often reduces to the identity. The assumption it has 0 dof is known as the ``counting argument''. However, it can be very difficult to spot isotropies, so care is needed: