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 |
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position |
direction |
orientation |
distance |
angulus |
parallelism |
Colinearity of biratio |
identity |
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+ |
+ |
+ |
+ |
+ |
+ |
translations |
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+ |
+ |
+ |
+ |
+ |
+ |
displacings |
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+ |
+ |
+ |
+ |
+ |
isometries |
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+ |
+ |
+ |
+ |
similarities |
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+ |
+ |
+ |
affine group |
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+ |
+ |
projective group |
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+ |
Soruce: nfm |
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