Table 1.

Different transformation groups and some properties they leave invariant.

Object property	Isometry group	Similarity group	Affine group	Projective group	Topological group
position	variable	variable	variable	variable	variable
orientation	variable	variable	variable	variable	variable
lengths	invariant	variable	variable	variable	variable
length ratios	invariant	invariant	variable	variable	variable
angles	invariant	invariant	variable	variable	variable
parallelism	invariant	invariant	invariant	variable	variable
parallel length ratios	invariant	invariant	invariant	variable	variable
colinearity	invariant	invariant	invariant	invariant	variable
cross-ratio s	invariant	invariant	invariant	invariant	variable
coincidence	invariant	invariant	invariant	invariant	invariant
closure	invariant	invariant	invariant	invariant	invariant
number of holes	invariant	invariant	invariant	invariant	invariant