Geometric Transformations and Invariants 

Group 
Structure of invariant 
Set of transformation 
Bijective transformations on functions 
Collineation 
Conservation of the property to be a line 
Dilatations 
Collineation that maintain parallelism 
Similarities 
Multiply distances by a constant factor 
Isometries 
Keep all distances 
Displacements 
Keep all distances and orientation 
Translations 
Keep all distances orientation and each acts as an identical transformation on a family of parallels containing all the points 
Trivial group 
Just one element: identity 
Invariant of main subgroups of objective group 


position 
direction 
orientation 
distance 
angulus 
parallelism 
Colinearity of biratio 
identity 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
translations 

+ 
+ 
+ 
+ 
+ 
+ 
displacings 


+ 
+ 
+ 
+ 
+ 
isometries 



+ 
+ 
+ 
+ 
similarities 




+ 
+ 
+ 
affine group 





+ 
+ 
projective group 






+ 
Soruce: nfm 