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 :
Theorem: The number of independant invariants of a configuration under transformations is
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: