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]:
Theorem: The number of independant invariants of a configuration
Consider a few examples:
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: