Geometrical transformation

One calls geometrical transformation , all Bijection of part of a geometrical whole in itself.

One can try one or of classifications of these transformations.

Initially according to the dimension of the geometrical unit; one will thus distinguish mainly the plane transformations and the transformations in space.

One can also classify the transformations according to their preserved elements:

the Isométrie S, preserving the Distance S
the similarities, preserving the reports/ratios distances
the transformations closely connected, preserving the lines and their parallelism
the homographic transformations, preserving the lines

Each one as of these classes contains the preceding one.

the circular transformations, preserving the whole of the right-hand sides and the circles in the plane case, or transformations of Moebius , preserving the whole of the plans and the spheres, in dimension 3.

the transformations bidifférentiables or Difféomorphisme S are the transformations which are closely connected with the first order; they contain the preceding ones like particular cases, but also:

the transformations in conformity or anticonformes, preserving the angles, which are, at the first order, of the similarities

the equivalent transformations or équiaréales, preserving the surfaces in the plane case, or volumes in the case 3D, which is, with the first order, of the transformations closely connected of determinant 1

And finally, including the preceding ones:

the transformations bicontinues or homeomorphisms, preserving the vicinities of the points

groups and sub-groups of transformations are then created.

The study of the Géométrie is mainly the study of these transformations.

Nonexhaustive classification of the transformations according to their degree of complexity

the reflections according to a line (in the plan or space) or according to a plan (in space)
the symmetry S power stations
the translation S
the Rotation S of center C (in the plan) or of axis (D) in space

the reflections, symmetries, translations, rotations are examples of isométries of the plan or space. Some preserve the directed angles and are then called Déplacement S. the whole of displacements forms a group.

homotheties

the homotheties and the isométries are examples of similarities plan or space. One even shows that these transformations generate the whole of the similarities. The similarities preserving the directed angles form a group called the group of the direct similarities.

affinities

affinities and the similarities are examples of transformations closely connected of the plan or space. One even shows that these transformations generate the whole of the transformations closely connected.

There exist also transformations which are not defined in the entire plan or space. Among those one can quote the inversions, the homologies which are homographic transformations

Random links: